Fix overlay showing "Idle" after browser refresh when game is running

[Copilot]

On WebSocket reconnect (e.g. OBS browser source refresh), the overlay had no state to render and defaulted to "Idle" — even with an active game — because the server only pushed state on game events and sent nothing to newly connected clients.

Changes

• overlay/server.py
  • Added _last_state: dict[str, Any] | None cache to OverlayServer
  • broadcast() now stores state before fanning out to clients
  • _websocket_endpoint() immediately sends _last_state to each new client on connect, if a state has been broadcast before

async def _websocket_endpoint(self, websocket: WebSocket) -> None:
    await websocket.accept()
    self._clients.add(websocket)
    if self._last_state is not None:
        await websocket.send_text(json.dumps(self._last_state))
    ...

• tests/test_overlay.py
  • Added test: client connecting after a broadcast immediately receives cached state
  • Added test: client connecting before any broadcast receives no spurious message

Original prompt

---

This section details on the original issue you should resolve

<issue_title>Game appears as "Idle" in overlay after a refresh enven it is currently running</issue_title>
<issue_description></issue_description>

Comments on the Issue (you are @copilot in this section)

• Fixes Game appears as "Idle" in overlay after a refresh enven it is currently running #37

---

💬 Send tasks to Copilot coding agent from Slack and Teams to turn conversations into code. Copilot posts an update in your thread when it's finished.