Extract mutable state from game.ts into a dedicated GameModel class to clarify data flow and state transitions.
Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)
Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>
Replace the ad-hoc StoneState | {...} union in drawStone/draw with a named DrawableStone type.
Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)
Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>
Replace std::sync::Mutex with tokio::sync::Mutex; extract try_join_room, register_player, spawn_forwarder, broadcast_room_state, spawn_message_handler, remove_player.
Use Game::process_throw in the message handler.
Ultraworked with [Sisyphus](https://github.com/code-yeongyu/oh-my-openagent)
Co-authored-by: Sisyphus <clio-agent@sisyphuslabs.ai>
