Im very new to flutter. Now im tring to make my original app and I want it to have riverpod system.
The first thing I faced is Google signIn, I want to use riverpod to use google SignIn.
This is my auth class… and Im wondering how to change this into code with riverpod implemented. I googled and chatted with GPT, so I assume how to do this... but I would like to have pro coders advice.

This is what I`ve done so far.

import 'package:flutter/material.dart';
import 'package:google_sign_in/google_sign_in.dart';
import 'package:riverpod_annotation/riverpod_annotation.dart';
import '../../pages/signin_page.dart';

part 'auth_service.g.dart';

@Riverpod()
class AuthService extends _$AuthService{
  User? user;
  final FirebaseAuth _auth = FirebaseAuth.instance;
  final GoogleSignIn _googleSignIn = GoogleSignIn();

  Future<void> signInWithGoogle() async {
    try {
      final GoogleSignInAccount? googleSignInAccount =
          await _googleSignIn.signIn();
      if (googleSignInAccount == null) {
        // 사용자가 Google 로그인 창을 닫거나 취소한 경우 처리
        return;
      }
      final GoogleSignInAuthentication googleSignInAuthentication =
          await googleSignInAccount.authentication;

      final AuthCredential credential = GoogleAuthProvider.credential(
        accessToken: googleSignInAuthentication.accessToken,
        idToken: googleSignInAuthentication.idToken,
      );

      final UserCredential userCredential =
          await FirebaseAuth.instance.signInWithCredential(credential);
      final User? user = userCredential.user;
      if (user != null) {
        print('로그인 성공: ${user.displayName} (${user.email})');
        // 로그인 후 추가 작업 수행
      }
      // Use the user object for further operations or navigate to a new screen.
    } catch (e) {
      print('The error occured : {$e.toString()}');
    }
  }


  //I think this is enough.
  Future<void> signOutAndNavigate(BuildContext context) async {
    try {
      await _googleSignIn.signOut();
      await _auth.signOut();
      if (context.mounted) {
        Navigator.of(context).pushReplacement(
          MaterialPageRoute(builder: (context) => const SigninPage()),
        );
      }
    } catch (e) {
      print('The error occured : {$e.toString()}');
      // 선택적으로 사용자 친화적인 오류 메시지를 표시합니다.
    }
  }
}

I watched many videos about riverpod, but I couldn`t find exact one.
How to use Riverpod code generator is just that ‘Add part 000.g.dart’, annotation(@riverpod), add extends _$000.