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

apps/backend/.env.example

@@ -12,3 +12,7 @@ SMTP_PORT=587
 SMTP_USER=user@example.com
 SMTP_PASS=your-smtp-password
 SMTP_FROM=Playzer <no-reply@example.com>
+
+# Google OAuth
+GOOGLE_CLIENT_ID=your-google-client-id
+GOOGLE_CLIENT_SECRET=your-google-client-secret

apps/backend/src/lib/auth.ts

@@ -17,7 +17,21 @@ export const auth = betterAuth({
   emailAndPassword: {
     enabled: true,
   },
-  trustedOrigins: [process.env.APP_BASE_URL ?? 'http://localhost:5173'],
+  socialProviders: {
+    google: {
+      clientId: process.env.GOOGLE_CLIENT_ID!,
+      clientSecret: process.env.GOOGLE_CLIENT_SECRET!,
+    },
+  },
+  advanced: {
+    generateUsername: async (data) => {
+      return `user_${Date.now()}`;
+    },
+  },
+  trustedOrigins: [
+    process.env.APP_BASE_URL ?? 'http://localhost:5173',
+    process.env.BETTER_AUTH_URL ?? 'http://localhost:3000',
+  ],
   plugins: [openAPI()],
 });