flutter How to persist favorite list file using Shared_preferences?

Question

I don't have an error but the data is not persisted maybe I'm using it in the wrong way please take some time to help me. please.!

  @override
            Widget build(BuildContext context) {
     return Scaffold(
  backgroundColor: const Color(0xFFF6F4ED),
  appBar: AppBar(
    actions: [
      Badge(
        badgeContent: Text('${savedWords.length}'),
        toAnimate: true,
        position: const BadgePosition(top: 0, end: 0),
        child: IconButton(
          icon: const Icon(Icons.bookmark),
          onPressed: () => pushToFavoriteWordsRoute(context),
        ),
      ),
    ],
    centerTitle: true,
    title: const Text('Documents'),
  ),
  body: loading
      ? const Center(child: CircularProgressIndicator())
      : _exfilePath.isEmpty
          ? const Center(child: Text('No documents...'))
          : GridView.builder(
              shrinkWrap: true,
              gridDelegate: const SliverGridDelegateWithFixedCrossAxisCount(
                crossAxisCount: 2,
              ),
              itemCount: _exfilePath.length,
              itemBuilder: (context, int index) {
                var name = _exPath[index];
                var fileIndex = _exfilePath[index];
                bool isSave = savedWords.contains(fileIndex);
                return Padding(
                  padding: const EdgeInsets.all(8.0),
                  child: GridTile(
                    header: Align(
                      alignment: Alignment.topRight,
                      child: PopupMenuButton<IconTextMenu>(
                        color: Colors.purple[50],
                        shape: RoundedRectangleBorder(
                            borderRadius: BorderRadius.circular(16)),
                        onSelected: (value)  async {
                          switch (value) {
                            case IconMenu.favorite:
                              var persistFile =
                                  await getFileListPreference(fileIndex);
                              setState(() {
                                if (isSave) {
                                  savedWords.remove(persistFile);
                                } else {
                                  savedWords.add(persistFile);
                                }
                              });

                              break;
                            case IconMenu.delete:
                              _exfilePath[index].delete();
                              refresh();

                              break;
                            case IconMenu.share:
                              break;
                            default:
                          }
                        },
                        itemBuilder: (context) => IconMenu.items
                            .map((e) => PopupMenuItem<IconTextMenu>(
                                value: e,
                                child: ListTile(
                                    leading: Icon(
                                      e.icon,
                                      color: isSave
                                          ? Colors.red
                                          : Colors.blueAccent,
                                    ),
                                    title: Text(e.text))))
                            .toList(),

here are the functions I use to store the data

               Future<File> setFileListPreference(File file) async {
         File f = File('');
        _prefs = await SharedPreferences.getInstance();

       for (File file in savedWords) {
       String stringfile = file.path;
      _prefs.setString(savepdf, stringfile);
      File convertToFile = File(stringfile);
     f = convertToFile;
    }
     return f = file;
     }

// GET FILE

       Future<File> getFileListPreference(File file) async {
     _prefs = await SharedPreferences.getInstance();
     _prefs.getString(savepdf);
 
     return file;
    }

your support will be very useful to me thanks in advance.

error: type 'List<Object?>' is not a subtype of type 'String?' in type cast