feat: integrate Google OAuth authentication provider and add login UI button

AGENTS.md

@@ -88,7 +88,26 @@ Biome is used (not ESLint). Config in `biome.json`:
 - `BETTER_AUTH_SECRET`, `BETTER_AUTH_URL` - Auth core
 - `APP_BASE_URL` - Frontend URL for trusted origins
 - `CORS_ORIGIN` - Frontend URL for CORS policy
+- `GOOGLE_CLIENT_ID`, `GOOGLE_CLIENT_SECRET` - Google OAuth (see below)
 
 **Frontend** (`apps/frontend/.env`):
 - `VITE_*` prefix required (Vite embeds these at build time)
 - `VITE_API_BASE_URL` - Backend URL (default: http://localhost:3000)
+
+## Google OAuth
+
+To enable login with Google:
+
+1. Create OAuth credentials in Google Cloud Console:
+   - Application type: Web application
+   - Authorized redirect URI: `{BETTER_AUTH_URL}/api/auth/callback/google`
+
+2. Add env vars to `.env.example`:
+   ```
+   GOOGLE_CLIENT_ID=your-google-client-id
+   GOOGLE_CLIENT_SECRET=your-google-client-secret
+   ```
+
+3. The backend configures `socialProviders.google` automatically.
+
+4. Frontend uses `authClient.signIn.social({ provider: 'google', callbackURL: 'http://localhost:5173' })`.