src/fourier.ipynb

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    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"warn\">\n",
    "\n",
    "<b>SageMath Kernel</b>\n",
    "\n",
    "In order to execute the cells in this notebook, you will have to use a SageMath kernel (tested with SageMath 10.2).\n",
    "<br>See **the README for more information on how to install it**.\n",
    "<br><br>Note that I'd love to upload the notebook to CoCalc, such that you don't have to install anything on your own.\n",
    "<br>However, this will only be possible once issues [#7560](https://github.com/sagemathinc/cocalc/issues/7560) and [#7608](https://github.com/sagemathinc/cocalc/issues/7608) are resolved.\n",
    "<br><br>\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"warn\">\n",
    "\n",
    "<b>Custom CSS styling</b>\n",
    "\n",
    "This notebook uses custom CSS styling. If you don't see this text on a colored background make sure to _not_ view it on GitHub, but instead in VSCode (or another editor that can render Jupyter notebooks with custom CSS). Note that the custom CSS also does not work in CoCalc due to a bug reported [here](https://github.com/sagemathinc/cocalc/issues/7560#issue-2302223105). In CoCalc, interactive audio also does not work as reported [here](https://github.com/sagemathinc/cocalc/issues/7608).\n",
    "<br><br>\n",
    "\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# **The basics of Fourier analysis**\n",
    "**_A project by Dominic Plein ([@Splience](https://youtube.com/@splience)) & Felix Lentze_**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In this project, we will discuss what sound is, how to represent it mathematically & in a computer, and how to examine its frequency content using Fourier analysis. We will use SageMath along the way for the implementation and for interactive plots. [SageMath](https://www.sagemath.org/) is a free open-source computer algebra system as superset of Python which bundles many great libraries like `numpy`, `scipy`, `matplotlib`, and more. In the frequency domain, we can manipulate the frequency spectrum of a sound as will be demonstrated by implementing a rudimentary low-pass filter to cut off high frequencies of a square wave signal.\n",
    "\n",
    "This projected is intended for educational purposes, i.e. we don't focus on performance of our code (for that, see the FFT algorithm implemented in system-level languages like Rust or C++). Basic knowledge of linear algebra as well as Python are required. The key points are explained next to the code. The reader is encouraged to experiment with the code and the provided sound samples.\n",
    "\n",
    "Throughout your journey of exploring the amazing world of Fourier in this notebook, you might want to have **the amazing book [Linear algebra, signal proceessing, and wavelets. A unified approach](https://www.uio.no/studier/emner/matnat/math/nedlagte-emner/MAT-INF2360/v15/kompendium/) by Øyvind Ryan** open next to you. When we reference a theorem/definition from this book, we refer to the January 21, 2015 Python edition."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### **🌟 References**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- Our main source of inspiration and reference for this document is the awesome book: [**\"Linear algebra, signal processing, and wavelets. A unified approach\"** by Øyvind Ryan. Version: January 21, 2015)](https://www.uio.no/studier/emner/matnat/math/nedlagte-emner/MAT-INF2360/v15/kompendium/applinalgpython.pdf). Multiple versions of this dcoument are available [here](https://www.uio.no/studier/emner/matnat/math/nedlagte-emner/MAT-INF2360/v15/kompendium/). We highly encourage the reader to read through this document and do the exercises provided there to get a depper understanding of what is covered here only in a brief manner. For our definitions and theorems, we mainly refer to this book.\n",
    "- We've also used the book [Linear Algebra and its application by David Lay et al.](https://archive.org/details/linear-algebra-its-applications-6th/page/400/mode/2up) for some additional insights, especially with regards to orthogonal spaces."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### **🌟 Preface: SageMath vs. Python?**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Just as a small preface (that you may skip), we want to discuss the relation between SageMath and Python as we will use SageMath for the implementation. On the [SageMath website](https://www.sagemath.org/library-press.html), we find this:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> Leverages Existing Software: SageMath does not reinvent the wheel for every known calculation. When possible, it uses existing tools to solve the problem and combine all of them in one unique interface. This concept not only exposes software packages to a wider audience, but also helps to increase the quality by submitting bugs upstream.\n",
    "\n",
    "> SageMath uses Python as its \"glue language\" to interface with all its components. Python is also SageMath's primary interface language and hence SageMath does not invent a new programming language as other mathematical software systems do. Python is well established among research communities and makes interfacing even less complicated.\n",
    "\n",
    "So, SageMath uses Python as \"glue language\". In fact, it provides a unified Python interface to many great libraries in the scientific community, among other: `numpy`, `scipy`, `matplotlib`, `Maxima`, `BLAS` etc. (see also [here](https://doc.sagemath.org/html/en/faq/faq-general.html#why-did-you-write-sage-from-scratch-instead-of-using-other-existing-software-and-or-libraries))."
   ]
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   "cell_type": "markdown",
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   "source": [
    "But then, you might wonder why you are able to write something like this in SageMath:\n",
    "\n",
    "```python\n",
    "f(x) = x^2\n",
    "```\n",
    "\n",
    "This is not valid python syntax: you cannot have a variable called `f(x)` in Python. And in order to raise a number to a power, you use `**` instead of `^`. In SageMath, however, you can do this. This is because SageMath is a superset of Python, i.e. it extends the syntax of Python with some niceties like this one to make your life easier. Technically, Sage includes a [preparsing](https://doc.sagemath.org/html/en/developer/coding_in_python.html#sage-preparsing) step to convert the special syntax to valid Python code. If you want to dive deeper, [this is the Python method](https://github.com/sagemath/sage/blob/ffbbea9cb2360340f1c5517b590883fdaf02575c/src/sage/repl/preparse.py#L1382) responsible to parse statements like the one above."
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     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Via SageMath syntax: 25\n",
      "Directly via Python (this is what SageMath does under the hood): 25\n"
     ]
    }
   ],
   "source": [
    "# Only possible in Sage (but just syntactic sugar)\n",
    "f(x) = x^2\n",
    "print(f\"Via SageMath syntax: {f(5)}\")\n",
    "\n",
    "# Sage will convert the above snippet to the following code internally.\n",
    "# You can find this out by executing `preparse(\"f(x) = x^2\")` in SageMath.\n",
    "__tmp__ = var(\"x\");\n",
    "f = symbolic_expression(x^Integer(2)).function(x)\n",
    "print(f\"Directly via Python (this is what SageMath does under the hood): {f(5)}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "SageMath is a superset of Python. Therefore, every Python code you write is also SageMath code by definition (but not the other way around as seen in the example). SageMath bundles many great open-source libraries and provides a unified interface to them."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### **🌟 Import packages**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "From the [Sage documentation](https://doc.sagemath.org/html/en/thematic_tutorials/numerical_sage/numerical_tools.html): <small>(highlights & line breaks by us)</small>\n",
    "\n",
    "> Sage has many different components that may be useful for numerical analysis. In particular three packages deserve mention, they are numpy, SciPy [...].\n",
    "> - **Numpy** is an excellent package that provides fast array facilities to python. It includes some basic linear algebra routines, vectorized math routines, random number generators, etc. It supports a programming style similar to one would use in matlab and most matlab techniques have an analogue in numpy.\n",
    "> - **SciPy** builds on numpy and provides many different packages for optimization, root finding, statistics, linear algebra, interpolation, FFT and dsp tools, etc."
   ]
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  {
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   "execution_count": 2,
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   "source": [
    "# Packages that ship with SageMath\n",
    "# see https://doc.sagemath.org/html/en/reference/spkg/ for all packages\n",
    "# that come distributed with Sage\n",
    "from scipy.io import wavfile\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Play audio samples in a Jupyter Notebook\n",
    "from IPython.display import Audio"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<hr>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### **🌟 Intro: What is sound \\& some first experiments**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We hear sounds every day - the chirping of birds, the rustling of leaves, music, people talking. Physically speaking, sound is a pressure wave that propagates through a medium like air. Variations in air pressure near our ears cause our eardrums to vibrate. Through some more processes, these vibrations are converted into electrical signals that are sent to our brain, which interprets these signals as sound. The frequency (number of variations per seconds) of these pressure waves determines the pitch of the sound we hear. The amplitude of the pressure wave determines the loudness of the sound. As an example, consider the following sine wave and experiment with its parameters to see how they affect the sound:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$$\n",
    "\\boxed{\n",
    "    A: \\mathbb{R} \\rightarrow \\mathbb{R}, \\quad\n",
    "    t \\mapsto A(t) \\coloneqq A_0 \\cdot \\sin(2\\pi f \\cdot t + \\varphi)\n",
    "}\n",
    "$$\n",
    "\n",
    "$$\n",
    "A_0: \\text{Amplitude}, \\quad\n",
    "f: \\text{Frequency}, \\quad\n",
    "\\varphi: \\text{Phase}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Watch out for your ears when playing the sound!**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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   "source": [
    "# Don't worry about the sample rate yet, we'll cover what it is later\n",
    "# when we talk about discrete signals.\n",
    "SAMPLE_RATE = float(25000)  # sample rate (samples per second, Hz)\n",
    "T_sound = 1  # duration of sound (seconds)"
   ]
  },
  {
   "cell_type": "markdown",
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    "<style>\n",
    ".cell-output-ipywidget-background {\n",
    "    background-color: transparent !important;\n",
    "}\n",
    ":root {\n",
    "    --jp-widgets-color: var(--vscode-editor-foreground);\n",
    "    --jp-widgets-font-size: var(--vscode-editor-font-size);\n",
    "}  \n",
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      "text/plain": [
       "Interactive function <function sin_wave at 0x7f0d7e5d3060> with 3 widgets\n",
       "  f: TransformIntSlider(value=440, description='f', max=1400)\n",
       "  phi: TransformFloatSlider(value=0.0, description='phi', max=6.283185307179586)\n",
       "  A_0: TransformFloatSlider(value=0.8, description='A_0', max=1.0)"
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   "source": [
    "@interact\n",
    "def sin_wave(f=slider(0, 1400, 1, default=440),\n",
    "        phi=slider(0,2*pi), A_0=slider(0,1,0.1, default=0.8)):\n",
    "\n",
    "    # Plot the sine wave (via SageMath)\n",
    "    A(t) = A_0 * np.sin(2*pi*f*t + phi)\n",
    "    show(plot(A(t), (t, 0, 0.06),\n",
    "            color=\"red\", thickness=1.5),\n",
    "            xmin=0, xmax=0.06, ymin=-1, ymax=1,\n",
    "            figsize=(10, 4), axes_labels=[\"t\", \"A(t)\"])\n",
    "\n",
    "    # Generate audio file\n",
    "    t_linspace = np.linspace(0, T_sound, int(T_sound*SAMPLE_RATE), endpoint=False)\n",
    "    wave = A_0 * np.sin(2*np.pi*f*t_linspace + phi)\n",
    "    display(Audio(wave, rate=SAMPLE_RATE, normalize=False))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This sine wave describes the variations of air pressure, which forms the tone we hear. The bigger the frequency $f$, the higher the pitch. The bigger the amplitude $A_0$, the louder the sound.\n",
    "\n",
    "It's therefore reasonable to say that we can model sounds by functions. A sound with frequency $f$ is modeled by a trigonometric function of that frequency. We call $\\sin(2\\pi f t)$ the **pure tone** of frequency $f$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>Observation (Ryan, 1.1).\n",
    "Sound as mathematical function.</strong>\n",
    "\n",
    "A sound can be represented by a mathematical function, with time as the free variable.<br>When a *function* represents a sound, it is often referred to as a *continuous sound*.\n",
    "<br><br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Summing up multiple sine waves**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "An interesting idea might be to sum up multiple sine waves of different frequencies. We will sum them up pointwise, i.e. add up their respective values at each point in time. Let's experiment with two sine waves with the same amplitude, but different frequencies:"
   ]
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       "Interactive function <function add_up_two_sin_waves at 0x7f0d7e5d2b60> with 2 widgets\n",
       "  f1: TransformIntSlider(value=540, description='f1', max=1000, min=400)\n",
       "  f2: TransformIntSlider(value=548, description='f2', max=1000, min=400)"
      ]
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   ],
   "source": [
    "@interact\n",
    "def add_up_two_sin_waves(\n",
    "    f1=slider(400, 1000, 1, default=540),\n",
    "    f2=slider(400, 1000, 1, default=548),\n",
    "):\n",
    "    def sound(t):\n",
    "        return np.sin(2 * np.pi * f1 * t) + np.sin(2 * np.pi * f2 * t)\n",
    "\n",
    "    # Plot square wave\n",
    "    t_for_plot = np.linspace(0, 0.04, 3000)\n",
    "    y = sound(t_for_plot)\n",
    "\n",
    "    plt.figure(figsize=(10, 4))\n",
    "    plt.plot(t_for_plot, y, color=\"red\", linewidth=1.5)\n",
    "    plt.xlabel(\"t\"); plt.ylabel(\"A(t)\")\n",
    "\n",
    "    # Generate audio file\n",
    "    T_sound = 2\n",
    "    t_for_audio = np.linspace(0, T_sound, int(T_sound * SAMPLE_RATE), endpoint=False)\n",
    "    wave = sound(t_for_audio)\n",
    "    display(Audio(wave, rate=SAMPLE_RATE))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A more complex example is the following one. This series approximates a square wave by adding up many individual sine waves (pure tones). It might seem quite surprising that we can indeed approximate a rectangular-looking wave by adding up \"roundish\" sine waves. We even add up discrete frequencies, but the sum of them approximates a continuous function. Listen to how the sound changes when we add more and more pure tones with higher frequencies.\n",
    "$$\n",
    "\\begin{align*}\n",
    "A(t) &\\coloneqq \\sum_{n=0}^{N}\n",
    "\\frac{4}{(2n+1)\\pi} \\sin\\left(2\\pi (2n+1) \\frac{t}{T}\\right)\\\\\n",
    "\n",
    "&= \\frac{4}{\\pi} \\sin\\left(2\\pi \\cdot \\frac{t}{T}\\right)\n",
    "+ \\frac{4}{3\\pi} \\sin\\left(2\\pi \\cdot 3 \\frac{t}{T}\\right)\n",
    "+ \\frac{4}{5\\pi} \\sin\\left(2\\pi \\cdot 5\\frac{t}{T}\\right)\n",
    "+ \\frac{4}{7\\pi} \\sin\\left(2\\pi \\cdot 7 \\frac{t}{T}\\right)\n",
    "+ \\cdots \n",
    "\\end{align*}\n",
    "$$"
   ]
  },
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       "Interactive function <function square_wave at 0x7f0d761e99e0> with 1 widget\n",
       "  N: Dropdown(description='N', options=(2, 4, 6, 8, 10, 20, 40, 80, 180, 1000), value=2)"
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   "source": [
    "@interact\n",
    "def square_wave(N=[2, 4, 6, 8, 10, 20, 40, 80, 180, 1000]):\n",
    "    def sound(t):\n",
    "        T = 0.5  # period of square wave: 2\n",
    "        f = 300  # fixed frequency [Hz]\n",
    "        y = np.zeros(np.shape(t))\n",
    "        for n in range(1, N, 2):\n",
    "            y = y + 4 / (n * np.pi) * np.sin(2 * np.pi * f * n * t / T)\n",
    "        return y\n",
    "\n",
    "    # Plot square wave\n",
    "    t_for_plot = np.linspace(0, 0.005, 1000)\n",
    "    y = sound(t_for_plot)\n",
    "\n",
    "    plt.figure(figsize=(10, 4))\n",
    "    plt.plot(t_for_plot, y, color=\"red\", linewidth=1.5)\n",
    "    plt.xlabel(\"t\"); plt.ylabel(\"A(t)\")\n",
    "\n",
    "    # Generate audio file\n",
    "    t_for_audio = np.linspace(0, T_sound, int(T_sound * SAMPLE_RATE), endpoint=False)\n",
    "    wave = sound(t_for_audio)\n",
    "    display(Audio(wave, rate=SAMPLE_RATE))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using another formula involving cosine functions, we can also approximate a triangle function. Compared to the square function it's a lot more pleasant to listen to; the sound is less \"harsh\". But not as \"smooth\" as for the pure sine wave.\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "A(t) &\\coloneqq \\sum_{n=0}^{N}\n",
    "\\frac{8}{(2n+1)^2\\pi^2} \\cos\\left(2\\pi (2n+1) \\frac{t}{T}\\right)\\\\\n",
    "&= - \\frac{8}{\\pi^2} \\cos\\left(2\\pi \\cdot \\frac{t}{T}\\right)\n",
    "- \\frac{8}{3^2\\pi^2} \\cos\\left(2\\pi \\cdot 3 \\frac{t}{T}\\right)\n",
    "- \\frac{8}{5^2\\pi^2} \\cos\\left(2\\pi \\cdot 5\\frac{t}{T}\\right)\n",
    "- \\frac{8}{7^2\\pi^2} \\cos\\left(2\\pi \\cdot 7 \\frac{t}{T}\\right)\n",
    "- \\cdots\n",
    "\\end{align*}\n",
    "$$"
   ]
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       "Interactive function <function triangle_wave at 0x7f0d748b8180> with 1 widget\n",
       "  N: Dropdown(description='N', options=(2, 4, 6, 8, 10, 20, 40, 80, 180, 1000), value=2)"
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   "source": [
    "@interact\n",
    "def triangle_wave(N=[2, 4, 6, 8, 10, 20, 40, 80, 180, 1000]):\n",
    "    def sound(t):\n",
    "        T = 0.5  # period of square wave: 2\n",
    "        f = 440  # fixed frequency [Hz]\n",
    "        y = np.zeros(np.shape(t))\n",
    "        for n in range(1, N, 2):\n",
    "            y = y + (-8) / (n**2 * np.pi**2) * np.cos(2 * np.pi * f * n * t / T)\n",
    "        return y\n",
    "\n",
    "    # Plot square wave\n",
    "    t_for_plot = np.linspace(0, 0.005, 1000)\n",
    "    y = sound(t_for_plot)\n",
    "\n",
    "    plt.figure(figsize=(10, 4))\n",
    "    plt.plot(t_for_plot, y, color=\"red\", linewidth=1.5)\n",
    "    plt.xlabel(\"t\")\n",
    "    plt.ylabel(\"A(t)\")\n",
    "\n",
    "    # Generate audio file\n",
    "    t_for_audio = np.linspace(0, T_sound, int(T_sound * SAMPLE_RATE), endpoint=False)\n",
    "    wave = sound(t_for_audio)\n",
    "    display(Audio(wave, rate=SAMPLE_RATE))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "At this point, we might ask ourselves, whether we can construct any arbitrary sound by summing up pure tones (sine & cosine waves). Our observation or rather our hope is that we can *de*compose any arbitrary sound into a sum of pure tones. This would give us a powerful tool to analyze and manipulate sounds as we could directly work with the different frequencies represented in a sound."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>A first hope (Ryan, 1.8).\n",
    "Decomposition of sound into pure tones.</strong>\n",
    "\n",
    "Any sound $f$ is a sum of pure tones at different frequencies. The amount of each frequency required to form $f$ is the *frequency content* of $f$. Any sound can be reconstructed from its frequency content.\n",
    "<br><br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### **🌟 Fourier series**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We've seen that we can identify a sound with a function. Now, we will make our first hope more precise. Enter the **Fourier analysis**, which allows us to model sound as a sum of frequency components. That is, we want to **approximate functions with the help up trigonometric functions**."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Periodicity**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the sake of simplicity, we will only consider periodic functions through this document."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"def\">\n",
    "\n",
    "<strong>Definition (Ryan, 1.5).\n",
    "Periodic functions.</strong>\n",
    "\n",
    "A function $f: \\mathbb{R} \\to \\mathbb{C}$ is *periodic* with period $T\\neq0$ if\n",
    "$$\n",
    "f(t) = f(t+T) \\quad \\forall t \\in \\mathbb{R}\n",
    "$$\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In other words: if we know $f$ for a single period, we know it for all times. This is a very strong property that will be exploited constantly in the following. Real-world sounds are aperiodic most of the time. However, we can think of such an aperiodic signal \"as the limit of a periodic signal as the period becomes arbitrarily large\" ~ [Alan V. Oppenheim et al. Signals and Systems, 2nd edition, p. 286](https://math.stackexchange.com/a/4851674/1083321). Don't worry about this too much, just accept that we will only work with periodic functions for now. (You may want to deal with aperiodic functions later once you're ready). Let's dive into the theory."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Square-integrable functions \\& inner product**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our goal is to approximate a periodic function $f: [0,T] \\to \\mathbb{C}$ by a linear combination of the trigonometric functions $\\sin$ and $\\cos$. The first question is: **How general do we allow $f$ to be?** From here on, we will demand our function $f$ to be square-integrable."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"def\">\n",
    "\n",
    "<strong>Definition (Ryan, 1.13).\n",
    "Continuous & square-integrable functions.</strong>\n",
    "\n",
    "The set of continuous functions defined on an interval $[0,T]$ is denoted $C[0,T]$.\n",
    "\n",
    "The function $f: [0,T] \\to \\mathbb{C}$ is said to be **square integrable** on $[0,T]$ if $f^2$ is Riemann-integrable,\n",
    "<br>i.e., if the Riemann integral of $f^2$ on $[0,T]$ exists:\n",
    "\n",
    "$$\n",
    "\\int_0^T \\bigl|f(t)\\bigr|^2 \\, \\text{d}t < \\infty\n",
    "$$\n",
    "\n",
    "The set of all square integrable functions on $[0,T]$ is denoted $L^2[0,T]$.\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<!-- In theory, we don't need the strong assumption $f \\in C[0,T]$. Instead, $f\\in L^2[0,T]$ (i.e. $f$ is square-integrable) would be sufficient. However, in practice, we will often work with continuous functions and for the sake of this seminar and to simplify the discussion, we will assume that $f \\in C[0,T]$.\n",
    "\n",
    "We can also define an inner product for the vector space $L[0,T]$: -->\n",
    "\n",
    "So when we write $f\\in L^2[0,T]$, we mean that $f$ is square integrable on the interval $[0,T]$. This ensures that the function is \"well-behaved\" in some sense. If you're not too sure what square-integrable functions really are, don't worry too much and just think of them as functions that are \"nice\" to work with. Such a function is the square wave function, which is not continuous (it has a discontinuity), but square-integrable on $[0,T]$.\n",
    "\n",
    "We can also define an inner product on $L^2[0,T]$:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"def\">\n",
    "\n",
    "<strong>Definition (Ryan, 1.14).\n",
    "Inner product.</strong>\n",
    "\n",
    "<!-- Both $L^2[0,T]$ and $C[0,T]$ are vector spaces.\n",
    "<br>If the two functions $f$ and $g$ lie in $L^2[0,T]$ (or in $C[0,T]$), then the product $fg$ is also in $L^2[0,T]$ (or in $C[0,T]$). -->\n",
    "\n",
    "<!-- Moreover, both spaces are inner product spaces, with inner product defined by -->\n",
    "\n",
    "The inner product for $L^2[0,T]$ is defined by:\n",
    "\n",
    "$$\n",
    "\\langle f,g \\rangle \\coloneqq \\frac{1}{T} \\int_0^T f(t) \\overline{g(t)} \\, \\text{d}t\n",
    "$$\n",
    "The associated norm is then given by:\n",
    "$$\n",
    "\\|f \\| \\coloneqq \\sqrt{\\langle f,f \\rangle} = \\sqrt{\\frac{1}{T} \\int_0^T \\bigl|f(t)\\bigr|^2 \\, \\text{d}t}\n",
    "$$\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Here, $\\overline{g(t)}$ is the complex conjugate of $g(t)$. The normalizing factor $\\frac{1}{T}$ is just needed to ensure the constant function $f(t) = 1$ has norm $1$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Basis \\& linear combination**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So you might ask: \"Why do we need the inner product?\" It plays a crucial role in the following.<br>\n",
    "First, we define a basis $\\mathscr{F}_{N,T}$ for a subspace $V_{N,T} \\subset L^2[0,T]$ that is spanned by $2N+1$ pure tone functions."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"def\">\n",
    "\n",
    "<strong>Definition (Ryan, 1.15/1.22).\n",
    "Complex Fourier basis.</strong>\n",
    "\n",
    "- We define the following set functions and call it the **Nth order complex Fourier basis** for $V_{N,T}$:\n",
    "\n",
    "$$\n",
    "\\mathscr{F}_{N,T} \\coloneqq \\{ \n",
    "    e^{-2\\pi i \\frac{N t}{T}},\n",
    "    e^{-2\\pi i \\frac{(N-1) t}{T}},\n",
    "    \\cdots,\n",
    "    e^{-2\\pi i \\frac{t}{T}},\n",
    "    1,\n",
    "    e^{2\\pi i \\frac{t}{T}},\n",
    "    \\cdots,\n",
    "    e^{2\\pi i \\frac{(N-1) t}{T}},\n",
    "    e^{2\\pi i \\frac{N t}{T}}\n",
    "\\}\n",
    "$$\n",
    "\n",
    "- $V_{N,T}$ is defined as the subspace of $L^2[0,T]$ spanned by the set of functions $\\mathscr{F}_{N,T}$.<br>$V_{N,T}$ is called the **Nth-order Fourier space**.\n",
    "\n",
    "- $f_N$ is defined as the best approximation of $f$ from $V_{N,T}$ with respect to the inner product defined above.<br>$f_N$ is called the **Nth-order Fourier series approximation of $f$**.\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "- One can verify that the functions $e^{2\\pi i \\frac{nt}{T}}$ are orthonormal with respect to the inner product defined above.\n",
    "- These functions are called \"pure tones\". They consist of cosine and sine functions (according to Euler's formula). Note that you can indeed also write the Fourier basis with $\\cos$ and $\\sin$ functions directly, but the complex exponential functions are more convenient to work with, so we'll stick with them.\n",
    "- $f_N$ is defined as the best approximation of $f$ from $V_{N,T}$ with respect to the inner product defined above.<br>Now what's a best approximation? Let's consult linear algebra:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>Theorem (Lay et al. Orthogonality \\& Least Squares, Theorem 9).\n",
    "Best approximation theorem.</strong>\n",
    "\n",
    "Let $W \\subseteq \\mathbb{R}^n$, let $y \\in \\mathbb{R}^n$, and let $\\hat{y}$ be the orthogonal projection of $y$ onto $W$.\n",
    "<br>Then $\\hat{y}$ is the closest point in $W$ to $y$, in the sense that\n",
    "\n",
    "$$\n",
    "\\| y - \\hat{y} \\| < \\| y - v \\| \\text{ for all } v \\in W \\text{ distinct from } \\hat{y}\n",
    "$$\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In our case, we don't consider $\\mathbb{R}^n$, but instead $L^2[0,T]$ with our norm associated to the inner product defined above."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since $f_N$ lives in $V_{N,T}$ (which is spanned by the basis $\\mathscr{F}_{N,T}$ of pure tones), we can write $f_N$ as a linear combination of the basis functions (per definition of a basis):\n",
    "\n",
    "$$\n",
    "f_N(t) = \\sum_{n=-N}^{N} z_n \\cdot e^{2\\pi i \\frac{n t}{T}}\n",
    "$$\n",
    "\n",
    "The $e^{2\\pi i \\frac{n t}{T}}$ are the basis functions (pure tones) and the $z_n$ are the coefficients of the linear combination.<br>Let's visualize the current situation:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "![Fourier series](../images/orthogonal%20decomposition.png)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The question is now: **How to calculate the coefficients $z_n$?** The following theorem gives an answer:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>Theorem (Ryan, 1.16/1.23).\n",
    "Fourier series coefficients.</strong>\n",
    "\n",
    "The set $\\mathscr{F}_{N,T}$ is an orthonormal basis for $V_{N,T}$. In particular, the dimension of $V_{N,T}$ is $2N+1$.\n",
    "<br>We denote by $z_{-N}, \\cdots, z_0, \\cdots, z_N$ the coordinates of $f_N$ in the basis $\\mathscr{F}_{N,T}$, i.e.\n",
    "\n",
    "$$\n",
    "f_N(t) = \\sum_{n=-N}^{N} z_n \\cdot e^{2\\pi i \\frac{n t}{T}}\n",
    "$$\n",
    "\n",
    "The $z_n$ are called the complex Fourier coefficients of $f$, and they are given by:\n",
    "\n",
    "$$\n",
    "z_n = \\langle f, e^{2\\pi i \\frac{n t}{T}} \\rangle\n",
    "= \\frac{1}{T} \\int_0^T f(t) \\cdot e^{-2\\pi i \\frac{n t}{T}} \\, \\text{d}t\n",
    "$$\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "So where does the formula for the complex numbers $z_n$ come from? Well, that's the orthogonal decomposition theorem in action:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>Theorem (Lay et al. Orthogonality \\& Least Squares, Theorem 9).\n",
    "Orthogonal Decomposition.</strong>\n",
    "\n",
    "Let $W$ be a subspace of $\\mathbb{R}^n$. Then each $y \\in \\mathbb{R}^n$ can be written uniquely in the form\n",
    "$$\n",
    "y = \\hat{y} + p \\quad \\text{ where } \\hat{y} \\in W \\text{ and } p\\in W^\\perp\n",
    "$$\n",
    "In fact, if $\\{u_1, \\cdots, u_p\\}$ is any orthogonal basis of $W$, then:\n",
    "$$\n",
    "\\hat{y} = \\frac{y\\cdot u_1}{u_1 \\cdot u_1} u_1 + \\cdots\n",
    "+ \\frac{y\\cdot u_p}{u_p \\cdot u_p} u_p\n",
    "$$\n",
    "and $p = y - \\hat{y}$.\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "You might be familiar with this in the context of the Gram-Schmidt process. Here again, we don't consider the space $\\mathbb{R}^n$, but instead $L^2[0,T]$. In our case, $W$ is the subspace $V_{N,T}$ spanned by the basis $\\mathscr{F}_{N,T}$ of pure tones. The orthogonal projection $\\hat{y}$ of $f$ onto $V_{N,T}$ is the best approximation $f_N$ of $f$ from $V_{N,T}$. Instead of the dot product in $\\mathbb{R}^n$, we use the inner product defined above. For $u_i$ we use our pure tone basis functions $e^{2\\pi i n t / T}$. Making use of $\\langle e^{2\\pi i n t / T}, e^{2\\pi i n t / T} \\rangle = 1$ (orthonormal), we obtain:\n",
    "\n",
    "$$\n",
    "\\begin{align*}\n",
    "f_N(t) &= \\sum_{n=-N}^{N} \\frac{\\langle f, e^{2\\pi i n t / T}\\rangle}\n",
    "{\\langle e^{2\\pi i n t / T}, e^{2\\pi i n t / T} \\rangle} e^{2\\pi i n t / T}\n",
    "= \\sum_{n=-N}^{N} \\langle f, e^{2\\pi i n t / T}\\rangle e^{2\\pi i n t / T}\\\\\n",
    "&= \\sum_{n=-N}^{N}\n",
    "\\underbrace{\\left( \\frac{1}{T} \\int_0^T f(t) e^{-2\\pi i n t / T} \\: \\text{d}t\\right)}_{\\eqqcolon z_n}\n",
    "e^{2\\pi i n t / T}\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is the formula for the Fourier series coefficients $z_n$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Convergence**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's not trivial to find conditions for $f$ that guarantee the convergence of its Fourier series. We won't go into detail here (if you're interested, search for \"Dirichlet conditions for the convergence of the Fourier series\").<br>Instead, it's sufficient for us to know that when we choose $N$ large enough, \"any reasonable periodic function can be approximated arbitrarily well by its $N$ th-order Fourier series approximation\" ~ [Ryan, p. 15]. That is, we will assume the following to be true for the rest of this document (it's not true in general, but good enough for our choice of functions):\n",
    "\n",
    "$$\n",
    "\\lim_{N\\to \\infty} f_N(t) = f(t) \\quad \\forall t\\in \\mathbb{R}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Solve integral with SageMath**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Using SageMath's [symbolic integration](https://doc.sagemath.org/html/en/reference/calculus/sage/symbolic/integration/integral.html), we are able to calculate the Fourier coefficients $z_n$ for the square wave we've considered beforehand. As can be seen above, the Fourier coefficients are given by:\n",
    "$$\n",
    "z_n = \\frac{1}{T} \\int_0^T f(t) \\cdot e^{-2\\pi i \\frac{n t}{T}} \\, \\text{d}t\n",
    "$$\n",
    "Note that they depend on $n$, so we define a function `frequency_content(n)` dependent on that parameter."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5636a133bf0b4ff79fd4dd34aaa819b9",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Interactive function <function show_frequency_content at 0x7f0d74921ee0> with 1 widget\n",
       "  n: TransformIntSlider(value=1, description='n', max=20, min=-20)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "var(\"t\")\n",
    "f = 440  # frequency [Hz]\n",
    "T = 1/f # period [s]\n",
    "square_wave(t) = sgn(sin(2 * pi * t / T))\n",
    "\n",
    "\n",
    "def frequency_content(n):\n",
    "    pure_tone(t) = e^(-2 * pi * i * n * t / T)\n",
    "    integrand = square_wave(t) * pure_tone(t)\n",
    "    return 1/T * integrand.integral(t, a=0, b=T)\n",
    "\n",
    "\n",
    "@interact\n",
    "def show_frequency_content(n=slider(-20, 20, 1, default=1)):\n",
    "    amount = frequency_content(n)\n",
    "    magnitude = abs(amount) if n != 0 else 0\n",
    "    print(f\"💫 How much is frequency {abs(n)}*{f}Hz = {n*f}Hz represented in signal: {magnitude}\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This is exactly how we found the Fourier coefficients above for the square wave and the triangle wave. For the square wave, we get:\n",
    "$$\n",
    "\\begin{align*}\n",
    "|A(t)| &\\coloneqq \\left|\\sum_{n=0}^{N}\n",
    "\\frac{4}{(2n+1)\\pi} \\sin\\left(2\\pi (2n+1) \\frac{t}{T}\\right)\\right|\\\\\n",
    "% &= \\left| \\frac{4}{\\pi} \\sin\\left(2\\pi \\cdot \\frac{t}{T}\\right)\n",
    "% + \\frac{4}{3\\pi} \\sin\\left(2\\pi \\cdot 3 \\frac{t}{T}\\right)\n",
    "% + \\frac{4}{5\\pi} \\sin\\left(2\\pi \\cdot 5\\frac{t}{T}\\right)\n",
    "% + \\frac{4}{7\\pi} \\sin\\left(2\\pi \\cdot 7 \\frac{t}{T}\\right)\n",
    "% + \\cdots\\right|\\\\\n",
    "&= \\frac{4}{\\pi} \\left| \\sin\\left(2\\pi \\cdot \\frac{t}{T}\\right)\\right|\n",
    "+ \\frac{4}{3\\pi} \\left| \\sin\\left(2\\pi \\cdot 3 \\frac{t}{T}\\right)\\right|\n",
    "+ \\frac{4}{5\\pi} \\left| \\sin\\left(2\\pi \\cdot 5\\frac{t}{T}\\right)\\right|\n",
    "+ \\frac{4}{7\\pi} \\left| \\sin\\left(2\\pi \\cdot 7 \\frac{t}{T}\\right)\\right|\n",
    "+ \\cdots\\\\\n",
    "&= \\frac{2}{\\pi} \\left|e^{2\\pi i \\frac{t}{T}} - e^{-2\\pi i \\frac{t}{T}}\\right|\n",
    "+ \\frac{2}{3\\pi} \\left|e^{2\\pi i \\cdot 3 \\frac{t}{T}} - e^{-2\\pi i \\cdot 3 \\frac{t}{T}}\\right|\n",
    "+ \\frac{2}{5\\pi} \\left|e^{2\\pi i \\cdot 5 \\frac{t}{T}} - e^{-2\\pi i \\cdot 5 \\frac{t}{T}}\\right|\n",
    "+ \\frac{2}{7\\pi} \\left|e^{2\\pi i \\cdot 7 \\frac{t}{T}} - e^{-2\\pi i \\cdot 7 \\frac{t}{T}}\\right|\n",
    "+ \\cdots\n",
    "\\end{align*}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that we've used the fact that $\\sin(t) = \\frac{1}{2i}\\left(e^{it} - e^{-it}\\right)$ here. This follows from Euler's formula: $e^{ix} = \\cos(x) + i\\sin(x)$."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Frequency diagram**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "A common representation of the frequency content of a sound is an amplitude-frequency diagram. This diagram shows how much different frequencies (on the $x$-axis) contribute to the sound (on the $y$-axis). That is, we plot the value of the Fourier coefficients $z_n$ for different $n$, which in turn correspond to the different frequencies (see the Fourier basis)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "N = 20\n",
    "frequencies = range(-N, N + 1)\n",
    "amplitudes = [float(abs(frequency_content(n))) if n != 0 else float(0.0) for n in frequencies]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the frequency amplitude spectrum (discrete)\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.stem(frequencies, amplitudes, basefmt=\" \")\n",
    "plt.xlabel(\"Frequency f in terms of n\")\n",
    "plt.ylabel(\"Magnitude (coefficient z_n)\")\n",
    "plt.title(\"Frequency Amplitude Spectrum\")\n",
    "plt.xticks(np.arange(-N, N + 1, 2))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### **🌟 Discrete Fourier Transform (DFT)**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> On computers and various kinds of media players, the sound is digital, meaning that it is represented by a large number of function values, and not by a function defined for all time instances. [Ryan, p. 41]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Sampling**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Instead of functions $f(t)$, we now consider vectors $x$ with entries representing the air pressure at different discrete time steps. The theory for Fourier analysis we've already developed so far can be adapted to this discrete setting (in that setting, proving the theorems we've shown is also easier). Above, we assumed that the functions were periodic. Here, we will assume that the vectors are periodic, i.e. we only need the values from the first period to reconstruct the whole signal."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"def\">\n",
    "\n",
    "<strong>Definition (Ryan, 2.1). Digital sound.</strong>\n",
    "\n",
    "A digital sound is a sequence $x = \\{ x_i \\}_{i=0}^{N-1}$ that corresponds to measurements of the air pressure of a sound $f$,\n",
    "<br>recorded at a fixed rate of $f_s$ (called *sampling frequency* or *sampling rate*) measurements per second:\n",
    "$$\n",
    "x_k = f\\left(\\frac{k}{f_s}\\right), \\quad \\text{for } k = 0, 1, \\ldots, N\n",
    "$$\n",
    "\n",
    "The measurements are often referred to as samples.\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Therefore in SageMath/Python, we represent a sound by an array of values, where each value corresponds to the measured air pressure at a given time.**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In the following example, we create a square wave (of frequency $f = 1 \\,\\text{Hz}$), plot it and overlay the discrete samples. We vary the sample rate interactively making use of the `@interact` decorator."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "f73b59edd05440d6ac7aea4e3ce5e59c",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Interactive function <function sampled_square_wave at 0x7f0d761eb380> with 1 widget\n",
       "  sample_rate: TransformIntSlider(value=7, description='sample_rate', max=30, min=1)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "@interact\n",
    "def sampled_square_wave(sample_rate=slider(1, 30, default=7)):\n",
    "    def square(t):  # Square Sound\n",
    "        return np.sign(np.sin(2 * np.pi * 1.0 * t))  # frequency = 1 Hz\n",
    "\n",
    "    # Discrete samples (discrete time steps where we sample the continuous signal)\n",
    "    t = np.arange(0, 3, 1 / sample_rate)\n",
    "    x = square(t)\n",
    "    plt.stem(t[1:], x[1:], basefmt=\" \", linefmt=\"red\")\n",
    "\n",
    "    plt.xlabel(\"t\")\n",
    "    plt.ylabel(\"A(t)\")\n",
    "    plt.yticks([-1, 0, 1])\n",
    "    plt.title(\"Sampled square wave\")"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As discussed, we don't know functions $f(t)$ describing the sound we hear in real life. Instead, the air pressure is measured at discrete time steps and stored in a vector/array. This process is called **sampling**. The values are then available in the form of `.wav`-files (they store some additional meta information, e.g. the sample rate used, the number of channels [left/right] etc., but also our data).\n",
    "\n",
    "We can import such `.wav` files into SageMath and analyze them. Don't worry about the following code block, it's just for the generation of the `.wav`. What is more relevant here is how to import a given `.wav` file back into SageMath."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "                <audio  controls=\"controls\" >\n",
       "                    <source 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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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type=\"audio/wav\" />\n",
       "                    Your browser does not support the audio element.\n",
       "                </audio>\n",
       "              "
      ],
      "text/plain": [
       "<IPython.lib.display.Audio object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Generate a square wave and store it as a .wav file\n",
    "# As amplitude, we use the maximum possible value for 16-bit integers\n",
    "def square(t):\n",
    "    amplitude = np.iinfo(np.int16).max\n",
    "    return amplitude * np.sign(np.sin(2 * np.pi * 440.0 * t))\n",
    "\n",
    "T_sound = 1\n",
    "sample_rate = 44100\n",
    "t_for_audio = np.linspace(0, T_sound, int(T_sound * sample_rate), endpoint=False)\n",
    "wave = square(t_for_audio)\n",
    "\n",
    "# Play/save audio\n",
    "display(Audio(wave, rate=float(sample_rate)))\n",
    "wavfile.write(\"../data/square_wave.wav\", sample_rate, wave.astype(np.int16))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "                <audio  controls=\"controls\" >\n",
       "                    <source 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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//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gAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYD/f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9/AYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgP9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//38BgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGA/3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//f/9//3//fwGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYABgAGAAYA=\" type=\"audio/wav\" />\n",
       "                    Your browser does not support the audio element.\n",
       "                </audio>\n",
       "              "
      ],
      "text/plain": [
       "<IPython.lib.display.Audio object>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Import the saved audio file and read its contents (& play it)\n",
    "sample_rate, wav_data = wavfile.read(\"../data/square_wave.wav\")\n",
    "display(Audio(wav_data, rate=sample_rate))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Discrete Fourier analysis**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For the transition to the discrete setting, we need to adapt the theory of Fourier analysis. First, the inner product is defined slightly differently (as a sum instead of an integral beforehand):"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"def\">\n",
    "\n",
    "<strong>Definition (Ryan, 2.11).\n",
    "Euclidean inner product and norm for complex vectors.</strong>\n",
    "\n",
    "For complex vectors of length $N$, the Euclidean inner product is given by\n",
    "$$\\langle x, y \\rangle = \\sum_{k=0}^{N-1} x_k \\overline{y_k}$$\n",
    "\n",
    "The associated norm is\n",
    "$$\\|x\\| = \\sqrt{\\sum_{k=0}^{N-1} |x_k|^2}$$\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> In the previous chapter we saw that, using a Fourier series, a function with period\n",
    "$T$ could be approximated by linear combinations of functions (the pure tones). This can be generalized to vectors, but then the pure tones must also be vectors. [Ryan, p. 49]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"def\">\n",
    "\n",
    "<strong>Definition (Ryan, 2.12). Discrete Fourier analysis.</strong>\n",
    "\n",
    "In Discrete Fourier analysis, a vector $x = (x_0, \\ldots, x_{N-1})$ is represented as a linear combination of the $N$ vectors\n",
    "<!-- $$\n",
    "\\varphi_n = \\frac{1}{\\sqrt{N}} \\left(1, e^{2\\pi i n / N}, e^{2\\pi i 2n / N}, \\ldots, e^{2\\pi i kn / N}, \\ldots, e^{2\\pi i n (N-1) / N} \\right)\n",
    "$$ -->\n",
    "$$\n",
    "\\varphi_n = \\frac{1}{\\sqrt{N}}\n",
    "\\begin{pmatrix}\n",
    "1\\\\\n",
    "e^{2\\pi i n / N}\\\\\n",
    "e^{2\\pi i 2n / N}\\\\\n",
    "\\vdots\\\\\n",
    "e^{2\\pi i kn / N}\\\\\n",
    "\\vdots\\\\\n",
    "e^{2\\pi i n (N-1) / N}\n",
    "\\end{pmatrix}\n",
    "$$\n",
    "\n",
    "These vectors are called the normalised complex exponentials, or the pure digital tones of order $N$. $n$ is also called frequency index. The whole collection $\\mathscr{F}_N = \\{ \\varphi_n \\}_{n=0}^{N-1}$ is called the $N$-point Fourier basis.\n",
    "\n",
    "<br>\n",
    "</div>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>Lemma (Ryan, 2.13). Orthonormal basis.</strong>\n",
    "\n",
    "The normalized complex exponentials $\\{ \\varphi_n \\}_{n=0}^{N-1}$ of order $N$ form an orthonormal basis in $\\mathbb{R}^N$.\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\"The focus in Discrete Fourier analysis is to change coordinates from the standard basis to the Fourier basis\" [Ryan, p. 51] as described by the following theorem:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>Theorem (Ryan, 2.14/2.16). Discrete Fourier Transform.</strong>\n",
    "\n",
    "We will denote the change of coordinates matrix from the standard basis of $\\mathbb{R}^N$ to the Fourier basis $\\mathscr{F}_N$ by $F_N$.\n",
    "\n",
    "The Fourier matrix $F_N$ is the unitary $N \\times N$ matrix with entries given by\n",
    "$$\n",
    "(F_N)_{nk} = \\frac{1}{\\sqrt{N}} e^{-2\\pi i nk / N},\n",
    "$$\n",
    "for $0 \\le n, k \\le N - 1$.\n",
    "The matrix ${\\sqrt{N}}F_N$ is also called the ($N$-point) discrete Fourier transform (DFT).\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Our implementation of the DFT is based on exactly this representation."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "def dft(x):\n",
    "    N = len(x)  # number of samples\n",
    "    n = np.arange(N)  # Vector of increasing natural Numbers from 0 to N-1\n",
    "    k = n.reshape((N, 1))  # Transpose n; k is 1xN vector\n",
    "    M = -2 * i * np.pi * n * k / N  # Symmetric N x N - matrix\n",
    "    F_N = np.exp(M)  # Calculate DFT-Matrix coefficients\n",
    "\n",
    "    return np.dot(F_N, x)  # Matrix multiplication\n",
    "\n",
    "\n",
    "def idft(x):\n",
    "    N = len(x)  # number of samples\n",
    "    n = np.arange(N)  # Vector of increasing natural Numbers from 0 to N\n",
    "    k = n.reshape((N, 1))  # Transpose n; k is 1xN vector\n",
    "    M = 2 * i * np.pi * n * k / N  # Symmetric N x N - matrix\n",
    "    F_Ninv = np.exp(M)\n",
    "\n",
    "    return np.dot(F_Ninv, x)  # Matrix multiplication"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The inverse discrete fourier (IDFT) transform is justified by the following theorem:"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong> Theorem (Ryan, 2.17). Inverse Discrete Fourier Transform.</strong>\n",
    "\n",
    "The Matrix $F_N$ is invertable. Its coeffiecients can be computed from\n",
    "$$\n",
    "(F_N^{-1})_{nk} = {\\sqrt{N}} e^{2\\pi i nk / N}\n",
    "$$\n",
    "\n",
    "The matrix $\\frac{1}{\\sqrt{N}}F_N^{-1}$ is called Inverse Discrete Fourier Transform (IDFT).\n",
    "\n",
    "<br>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **DFT vs. Fourier series**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We've already seen the Fourier coefficients $z_n$ for the square wave. Let's assume we've sampled such a square wave and stored the values in a vector $x$. We can now calculate the Fourier coefficients for the discrete square wave with our implementation and plot them in a amplitude-frequency diagram. One can show that for $N\\to \\infty$, i.e. infinitely many samples, we get the same Fourier coefficients as for the continuous square wave."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that for the sake of simplicity, we assume $f\\in V_{M,T}$, i.e. $f$ is equal to its Fourier series approximation $f_M$ (for some $M$)."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<link rel=\"stylesheet\" href=\"./styles.css\">\n",
    "<div class=\"theorem\">\n",
    "\n",
    "<strong>Proposition (Ryan, 2.24). Relation between Fourier coefficients and DFT coefficients.</strong>\n",
    "\n",
    "Let $N > 2M$, $f \\in V_{M,T}$, and let $\\mathbf{x} = \\{ f(kT/N) \\}_{k=0}^{N-1}$ be $N$ uniform samples from $f$ over $[0, T]$.\n",
    "<br>The Fourier coefficients $z_n$ of $f$ can be computed from\n",
    "$$\n",
    "(z_0, z_1, \\ldots, z_M, \\underbrace{0, \\ldots, 0}_{N-(2M+1)}, z_{-M}, z_{-M+1}, \\ldots, z_{-1}) = \\frac{1}{N} \\text{DFT}_N(\\mathbf{x})\n",
    "$$\n",
    "\n",
    "In particular, the total contribution in $f$ from frequency $n/T$, for $0\\leq n\\leq M$, is given by $y_n$ and $y_{N-n}$, where $y$ is the $\\text{DFT}$ of $x$.\n",
    "\n",
    "<br>\n",
    "</div>\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **The square wave as an example**"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "def square(t):\n",
    "    # Try to change the frequency to 2.0 (or another value)\n",
    "    # and see how the amplitude-frequency spectrum changes\n",
    "    # in the plot below.\n",
    "    return sgn(np.sin(2 * np.pi * 1.0 * t))\n",
    "\n",
    "\n",
    "# Define our signal at discrete time steps\n",
    "sample_rate = 1000\n",
    "t = np.arange(0, 2, 1 / sample_rate)  # discrete samples\n",
    "x = square(t)\n",
    "\n",
    "# Calculate the Discrete Fourier Transform of our signal\n",
    "dft_square = dft(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot amplitude-frequency spectrum\n",
    "frequencies = np.arange(0, 0.5 * sample_rate, sample_rate / len(x))\n",
    "amplitude_spectrum = np.abs(dft_square)\n",
    "\n",
    "plt.figure(figsize=(10, 5))\n",
    "plt.stem(frequencies, amplitude_spectrum[:len(frequencies)], basefmt=\" \")\n",
    "plt.title(\"Amplitude-Frequency Spectrum\")\n",
    "plt.xlabel(\"Frequency f (Hz)\")\n",
    "plt.ylabel(\"Amplitude A(f)\")\n",
    "plt.xlim(0, 20)\n",
    "plt.xticks(np.arange(0, 20, 1))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "As we can see, the Frequency-Spectrum is the same compared to what we obtained beforehand in the case where we didn't sample the square wave and worked directly with its known function representation!\n",
    "\n",
    "<sub>For the inclined reader, note that we only calculate the positive frequencies here. Earlier, we've also shown the amplitudes for the negative frequencies.</sub>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    " Let's check that the IDFT makes sense by applying it to the DFT of the square wave. Since $\\text{IDFT}(\\text{DFT}(x)) = x$, we expect to get back the original square wave. Here, we only use the real part of the DFT coefficients. The imaginary part encodes the phase information, which we don't consider here. Note that in general, we have to consider both the real and imaginary parts to fully reconstruct the original signal."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 640x480 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Perform the inverse DFT\n",
    "idft_square = idft(dft_square)\n",
    "real_part_idft_square = np.real(idft_square)\n",
    "\n",
    "# Plot sound\n",
    "t_linspace = np.linspace(0, 3, 1000)\n",
    "plt.scatter(t, real_part_idft_square, color=\"red\")\n",
    "plt.xlabel(\"t\")\n",
    "plt.ylabel(\"A(t)\")\n",
    "plt.title(\"Reconstructed square wave\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### **Low-pass filter**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "> The focus in Discrete Fourier analysis is to change coordinates from standard basis to the Fourier basis, performing some operations on this \"Fourier representation\", and then change coordinates back to the standard basis. [Ryan, p. 51]"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We'll do this in the form of a low-pass filter, i.e. only low frequencies in a sound will pass and the high frequencies above a certain **\"cutoff frequency\"** will just be cut off. This is a common operation in audio processing any you have probably heard this effect before in music (referred colloquially to as \"opening/closing the filter\").\n",
    "\n",
    "The filter is easy to implement as we have our DFT and IDFT ready to use. This is our plan:\n",
    "- Transform the sound to the frequency domain via the DFT.\n",
    "- Then set the amplitudes of all higher frequencies (above the cutoff frequency) to $0$.\n",
    "- Finally, transform back to the time domain via the IDFT to be able to play back the sound."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "def square(t):\n",
    "    # This time, we use a frequency that one can actually hear\n",
    "    return sgn(np.sin(2 * np.pi * 350.0 * t))\n",
    "\n",
    "# Define our signal at discrete time steps\n",
    "sample_rate = 5000\n",
    "t = np.arange(0, 1, 1 / sample_rate)  # discrete samples\n",
    "x = square(t)\n",
    "\n",
    "# Calculate the Discrete Fourier Transform of our signal\n",
    "dft_square = dft(x)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot the square wave in the time domain\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.stem(t, x, basefmt=\" \", linefmt=\"red\")\n",
    "plt.xlabel(\"t\")\n",
    "plt.ylabel(\"A(t)\")\n",
    "plt.xlim(0, 0.01)\n",
    "plt.title(\"Original square wave (sampled)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample rate is 5000 Hz\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot amplitude-frequency spectrum\n",
    "print(f\"Sample rate is {sample_rate} Hz\")\n",
    "frequencies = np.arange(0, 0.5 * sample_rate, sample_rate / len(x))\n",
    "amplitude_spectrum = np.abs(dft_square)\n",
    "\n",
    "plt.figure(figsize=(10, 5))\n",
    "plt.stem(frequencies, amplitude_spectrum[:len(frequencies)], basefmt=\" \")\n",
    "plt.title(\"Amplitude-Frequency Spectrum (of original square wave)\")\n",
    "plt.xlabel(\"Frequency f (Hz)\")\n",
    "plt.ylabel(\"Amplitude A(f)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note that the small values $\\neq 0$ are due to numerical imprecisions and we will just ignore them here."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "def low_pass_filter(dft_signal, cutoff_freq):\n",
    "    N = len(dft_signal)\n",
    "    dft_filtered = dft_signal.copy()\n",
    "\n",
    "    # You might expect something like this instead:\n",
    "    # dft_filtered[(cutoff_freq + 1):] = 0\n",
    "    # This won't work as we have to filter out the negative frequencies as well.\n",
    "    # We need e^(-2*pi*i*f) and e^(2*pi*i*f) to be zero for f > cutoff_freq\n",
    "    # so instead, we do this:\n",
    "    dft_filtered[cutoff_freq + 1 : N - cutoff_freq] = zero_vector(N - 2 * cutoff_freq - 1)\n",
    "\n",
    "    return dft_filtered\n",
    "\n",
    "\n",
    "# Filter with our low-pass filter\n",
    "filtered = low_pass_filter(dft_square, cutoff_freq=1500)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Sample rate is 5000 Hz\n"
     ]
    },
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x500 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Plot amplitude-frequency spectrum of filtered signal\n",
    "print(f\"Sample rate is {sample_rate} Hz\")\n",
    "frequencies = np.arange(0, 0.5 * sample_rate, sample_rate / len(x))\n",
    "amplitude_spectrum = np.abs(filtered)\n",
    "\n",
    "plt.figure(figsize=(10, 5))\n",
    "plt.stem(frequencies, amplitude_spectrum[: len(frequencies)], basefmt=\" \")\n",
    "plt.title(\"Amplitude-Frequency Spectrum (of filtered square wave)\")\n",
    "plt.xlabel(\"Frequency f (Hz)\")\n",
    "plt.ylabel(\"Amplitude A(f)\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Note how all frequencies above the cutoff frequency are set to $0$. Let's finally transform back to the time-domain and listen to the filtered sound."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [],
   "source": [
    "# Go back to the time domain\n",
    "filtered_signal = idft(filtered)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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",
      "text/plain": [
       "<Figure size 1000x400 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# Finally plot the filtered square wave in the time domain\n",
    "plt.figure(figsize=(10, 4))\n",
    "plt.stem(t, np.real(filtered_signal), basefmt=\" \", linefmt=\"red\")\n",
    "plt.xlabel(\"t\")\n",
    "plt.ylabel(\"A(t)\")\n",
    "plt.xlim(0, 0.01)\n",
    "plt.title(\"Filtered square wave\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Since we've cut off some high frequencies, the square wave loses its shape, especially its sharp corners. The lower the cutoff frequency, the more the shape will resemble a sine wave as we remove more and more high freqruencies. You can try this out in our interactive demo here. Note that all frequencies _above_ the cutoff frequency will be set to $0$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {
    "vscode": {
     "languageId": "python"
    }
   },
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "72a2a8397271499ba96fa4187fe8f5fe",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "Manual interactive function <function low_pass_filter_interactively at 0x7f0d761eb560> with 1 widget\n",
       "  cutoff_freq: TransformIntSlider(value=500, description='cutoff_freq', max=7499, min=350)"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "# If the interactive cell takes too long to execute, try to lower the sample rate.\n",
    "# This will make the results less pretty, but the computation faster.\n",
    "sample_rate = float(30000)\n",
    "max_freq = int(sample_rate / 4) - 1\n",
    "\n",
    "@interact\n",
    "def low_pass_filter_interactively(cutoff_freq=slider(350,max_freq, default=500),\n",
    "                                  auto_update=False):\n",
    "    print(f\"Sample rate is {sample_rate} Hz\")\n",
    "    print(f\"Frequencies above {cutoff_freq} Hz will be cut off\")\n",
    "\n",
    "    # Define our signal at discrete time steps\n",
    "    t = np.arange(0, 0.5, 1 / sample_rate)\n",
    "    amplitudes = square(t)\n",
    "\n",
    "    # Translate to DFT to go to frequency domain\n",
    "    # Note that we're using numpy's fast implementation of the Fourier transform\n",
    "    # which uses advanced algorithms. You can replace this call by our custom\n",
    "    # dft() and for idft() respectively, however, this will be a lot slower then.\n",
    "    dft_square = np.fft.fft(amplitudes)\n",
    "\n",
    "    # Filter with our low-pass filter in the frequency domain\n",
    "    filtered = low_pass_filter(dft_square, cutoff_freq)\n",
    "\n",
    "    # Go back to the time domain\n",
    "    filtered_signal = np.fft.ifft(filtered)\n",
    "\n",
    "    # Finally plot the filtered square wave in the time domain\n",
    "    plt.figure(figsize=(10, 4))\n",
    "    plt.plot(t, np.real(filtered_signal), color=\"red\", marker=\"o\", markersize=3)\n",
    "    plt.xlabel(\"t\")\n",
    "    plt.ylabel(\"A(t)\")\n",
    "    plt.xlim(0, 0.01)\n",
    "    plt.title(\"Filtered square wave\")\n",
    "    plt.show()\n",
    "\n",
    "    # Show audio file\n",
    "    display(Audio(np.real(filtered_signal), rate=sample_rate))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice how the audio sounds more and more like a pure sine wave as we remove more and more higher frequencies (drag the `cutoff_freq` all to the left). If we let through many frequencies, we will hear the characteristic square wave sound again with all its \"harmonics\" (high-frequency components).\n",
    "\n",
    "We also see [Gibbs phenomenon](https://en.wikipedia.org/wiki/Gibbs_phenomenon) in action here, which states that if $f(t)$ has a discontinuity, then the partial sums of the Fourier series will \"overshoot\" the true value of $f(x)$ at the top part of the discontinuity and undershoot the bottom part."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "---"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Congratulations if you made it this far 🙌 You've learned a lot about Fourier analysis and how to apply it to sound. We hope you enjoyed this interactive environment. If you have any feedback, feel free to open a new issue on GitHub or comment under the respective YouTube video. Note that we're students, so don't expect a quick response from our side. We're still happy to hear from you.\n",
    "\n",
    "There's still so much more to discover with regards to Fourier analysis and we only covered the very basics of it here. We highly recommend the book by Øyvind Ryan that we cited many times throughout this document. Scroll up to the top to see the references and links."
   ]
  }
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