remember last played boombox song

although this is a very minute feature, it would be nice if the boombox remembered the song which was last played between reboots and reconnects. this could be done by storing the last played song id in the player's profile.

Start by locating the boombox playback flow and the player's profile persistence entry point. Confirm how reconnects and reboots restore player state, then verify that the last played song ID is saved and restored in both cases.

Written by the indexing model from the issue text.