Show user activity and readiness status in rooms

Rooms should show whether users are active, idle, competing, spectating, waiting, or currently solving.

Feedback source:
- User position for what they're doing.
- Show if people are active.
- In users online, you should be able to challenge someone, or invite them to a room.

Acceptance ideas:
- Display per-user room state in the room UI.
- Distinguish at least active/idle/spectating/competing/waiting/skipping if available.
- Use status information to support better leader transfer and host controls.

Start by locating the room UI and the existing users-online, user-position, and room-state data flows. Trace how challenge or invitation actions, leader transfer, and host controls currently work. Done means the room displays the requested per-user states and uses available status information for the described controls.

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