How to replace "notifyListeners()" with "update()" using GetX in Flutter?

I am trying to modify the following code that uses stateful widgets and provider package to use GetX package and stateless widgets instead. In a part of code it specifies if the user is authenticated or not to show the home or auth screen. This is the auth_controller.dart code:

class Auth with ChangeNotifier {
  String? _token;
  DateTime? _expiryDate;
  String? _userId;
  Timer? _authTimer;

  bool get isAuth {
    return token != null;
  }

  String? get token {
    if (_expiryDate != null &&
        _expiryDate!.isAfter(DateTime.now()) &&
        _token != null) {
      return _token;
    }
    return null;
  }

  String? get userId {
    return _userId;
  }

  Future<void> _authenticate(
      String email, String password, String urlSegment) async {
    final host = Platform.isAndroid ? '10.0.2.2' : '127.0.0.1';

    final url = Uri.parse('http://$host:8080/api/$urlSegment');
    // final url = Uri.parse('http://10.0.2.2:8000/api/$urlSegment');
    // final url = Uri.parse('http://127.0.0.1:8000/api/$urlSegment');

    try {
      final http.Response response = await http.post(
        url,
        headers: {"Content-Type": "application/json"},
        body: json.encode(
          {
            'email': email,
            'password': password,

            //'returnSecureToken': true,
          },
        ),
      );

      final responseData = json.decode(response.body);
      if (responseData['error'] != null) {
        throw HttpException(responseData['error']['message']);
      } else {
      _token = responseData['idToken'];
      _userId = responseData['id'];
      _expiryDate = DateTime.now().add(
        Duration(
          seconds: responseData['expiresIn'],
        ),
      );
      }
      _autoLogout();
      notifyListeners();
      final prefs = await SharedPreferences.getInstance();
      final userData = json.encode(
        {
          'token': _token,
          'userId': _userId,
          'expiryDate': _expiryDate!.toIso8601String(),
        },
      );

      prefs.setString('userData', userData);
    } catch (error) {
      throw error;
    }
  }

  Future<void> signup(String email, String password) async {
    return _authenticate(email, password, 'register');
  }

  Future<void> login(String email, String password) async {
    return _authenticate(email, password, 'login');
  }

  Future<bool> tryAutoLogin() async {
    final prefs = await SharedPreferences.getInstance();
    if (!prefs.containsKey('userData')) {
      return false;
    }
    final Map<String, Object> extractedUserData = Map<String, Object>.from(
        json.decode(prefs.getString('userData') as String));
    final expiryDate =
        DateTime.parse(extractedUserData['expiryDate'] as String);

    if (expiryDate.isBefore(DateTime.now())) {
      return false;
    }
    _token = extractedUserData['token'] as String;
    _userId = extractedUserData['userId'] as String;
    _expiryDate = expiryDate;
    notifyListeners();
    _autoLogout();
    return true;
  }

  Future<void> logout() async {
    _token = null;
    _userId = null;
    _expiryDate = null;
    if (_authTimer != null) {
      _authTimer!.cancel();
      _authTimer = null;
    }
    notifyListeners();
    final prefs = await SharedPreferences.getInstance();
    // prefs.remove('userData');
    prefs.clear();
  }

  void _autoLogout() {
    if (_authTimer != null) {
      _authTimer!.cancel();
    }
    final timeToExpiry = _expiryDate!.difference(DateTime.now()).inSeconds;
    _authTimer = Timer(Duration(seconds: timeToExpiry), logout);
  }
}

and this is main.dart:

void main() => runApp(MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return MultiProvider(
      providers: [
        ChangeNotifierProvider.value(
          value: Auth(),
        ),
      ],
      child: Consumer<Auth>(
        builder: (ctx, auth, _) => MaterialApp(
          title: 'MyShop',
          theme: ThemeData(
            primarySwatch: Colors.purple,
            accentColor: Colors.deepOrange,
            fontFamily: 'Lato',
          ),
          home: auth.isAuth
              ? MainScreen()
              : FutureBuilder(
                  future: auth.tryAutoLogin(),
                  builder: (ctx, authResultSnapshot) =>
                      authResultSnapshot.connectionState ==
                              ConnectionState.waiting
                          ? SplashScreen()
                          : AuthScreen(),
                ),
          routes: {
            MainScreen.routeName: (ctx) => const MainScreen(),
            UserAccountScreen.routeName: (ctx) => const UserAccountScreen(),

          },
        ),
      ),
    );
  }

I tried to change the class to extend GetXController, also make some variables as observable by adding .obs at the end of them and listening to their changes by wrapping the listener part inside the Obx. But I don't know what should I do with isAuth variable? It seems my new code after modification can not update the isAuth state and it is always `false and this keeps the authentication page always up and no way to go into the application for users.

EDIT: This is the main.dart after modification:

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return GetMaterialApp(
      initialRoute: AuthController.instance.isAuth
          ? homeScreenRoute
          : authenticationScreenRoute,

      unknownRoute: GetPage(
          name: '/not-found',
          page: () => PageNotFound(),
          transition: Transition.fadeIn),
      getPages: [
        GetPage(
            name: rootRoute,
            page: () {
              return SiteLayout();
            }),
        GetPage(
            name: authenticationScreenRoute,
            page: () => const AuthenticationScreen()),
        GetPage(name: homeScreenRoute, page: () => HomeScreen()),
      ],
      debugShowCheckedModeBanner: false,
      title: 'BasicCode',
      theme: ThemeData(
        scaffoldBackgroundColor: light,
        textTheme: GoogleFonts.mulishTextTheme(Theme.of(context).textTheme)
            .apply(bodyColor: Colors.black),
        pageTransitionsTheme: const PageTransitionsTheme(builders: {
          TargetPlatform.iOS: FadeUpwardsPageTransitionsBuilder(),
          TargetPlatform.android: FadeUpwardsPageTransitionsBuilder(),
        }),
        primarySwatch: Colors.blue,
      ),
      // home: AuthenticationPage(),
    );
  }
}