feat: sign in/register with code in Flutter App

[sashatalalasha]

This pull request adds sign in/register with code functionality to the Flutter app.

Checklist

• I have read the contributing guidelines and signed the CLA.
• I have referenced an issue containing the design document if my change introduces a new feature.
• I have read the security policy.
• I confirm that this pull request does not address a security vulnerability.
  If this pull request addresses a security vulnerability,
  I confirm that I got approval (please contact [email protected]) from the maintainers to push the changes.
• I have added tests that prove my fix is effective or that my feature works.
• I have added the necessary documentation within the code base (if appropriate).

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