feat: integrate Better Auth and enhance authentication flow

- Added Better Auth integration to the web application, allowing for dual login support with both custom and Better Auth systems.
- Updated authentication client configuration to dynamically set the base URL based on the environment.
- Enhanced chat components to utilize user authentication status, improving user experience and security.
- Refactored various components to support Better Auth, including error handling and user identity management.
- Improved notification handling and user feedback mechanisms during authentication processes.

packages/backend/package.json

@@ -20,6 +20,7 @@
     "typescript": "catalog:"
   },
   "dependencies": {
+    "@convex-dev/better-auth": "^0.9.7",
     "@dicebear/avataaars": "^9.2.4",
     "convex": "catalog:",
     "nodemailer": "^7.0.10"