boardgameio/boardgame.io ? reverse-engineered prompt

Build me a JavaScript library for making turn based board games, where I can define simple move functions and the game state updates automatically. It should handle multiplayer sync, server state, storage, and player turns without me having to write networking code.

I want it to support things like game phases with different rules, a lobby for creating and joining matches, basic AI bots that can play, and logs so I can step back through previous game states. Make it easy to use with a plain JavaScript client, and also work nicely with React and React Native if I want to build a custom UI later.

Please include a few working example games so I can see how to use it, and set it up so I can run the project locally and test it right away. If you need to check current docs or best practices, go look them up online first.