feat: multiplayer slice — end to end working

WebSocket server:
- WS auth via Better Auth session on upgrade request
- Router with discriminated union dispatch and two-layer error handling
- In-memory connections map with broadcastToLobby
- Lobby handlers: join, leave, start
- Game handlers: answer, resolve round, end game, game:ready for state sync
- Shared game state store (LobbyGameStore interface + InMemory impl)
- Timer map separate from store for Valkey-readiness

REST API:
- POST /api/v1/lobbies — create lobby + add host as first player
- POST /api/v1/lobbies/:code/join — atomic join with capacity/status checks
- getLobbyWithPlayers added to model for id-based lookup

Frontend:
- WsClient class with typed on/off, connect/disconnect, isConnected
- WsProvider owns connection lifecycle (connect/disconnect/isConnected state)
- WsConnector component triggers connection at multiplayer layout mount
- Lobby waiting room: live player list, copyable code, host Start button
- Game view: reuses QuestionCard, game:ready on mount, round results
- MultiplayerScoreScreen: sorted scores, winner highlight, tie handling
- Vite proxy: /ws and /api proxied to localhost:3000 for dev cookie fix

Tests:
- lobbyService.test.ts: create, join, retry, idempotency, full lobby
- auth.test.ts: 401 reject, upgrade success, 500 on error
- router.test.ts: dispatch all message types, error handling
- vitest.config.ts: exclude dist folder

Fixes:
- server.ts: server.listen() instead of app.listen() for WS support
- StrictMode removed from main.tsx (incompatible with WS lifecycle)
- getLobbyWithPlayers(id) added for handleLobbyStart lookup

apps/api/src/ws/handlers/lobbyHandlers.ts

@@ -6,6 +6,7 @@ import {
   deleteLobby,
   removePlayer,
   updateLobbyStatus,
+  getLobbyByIdWithPlayers,
 } from "@lila/db";
 import {
   addConnection,
@@ -87,7 +88,7 @@ export const handleLobbyStart = async (
   user: User,
 ): Promise<void> => {
   // Load lobby and validate
-  const lobby = await getLobbyByCodeWithPlayers(msg.lobbyId);
+  const lobby = await getLobbyByIdWithPlayers(msg.lobbyId);
   if (!lobby) {
     throw new NotFoundError("Lobby not found");
   }