Replaces the previous mock-based authentication in client settings with FirebaseAuth for sign out functionality. Updates dependencies and imports accordingly, and adds firebase_auth to the settings package and firebase_core to the client app. Also includes minor Dart type improvements and dependency reordering.
Reorganized and clarified dependency sections in multiple pubspec.yaml files, grouping architecture and feature packages. Updated import statements in AuthRepositoryImpl for direct usage of types. Added missing dependencies to client authentication. Removed obsolete documentation and summary files from design_system and client home packages.
Relocated all app directories (client, design_system_viewer, staff) and their contents under the new 'apps/mobile' path. This change improves project organization and prepares for future platform-specific structuring.
