feat(api): add game:ready message for client state sync

- WsGameReadySchema added to shared schemas and WsClientMessageSchema
- handleGameReady sends current game:question directly to requesting
  client socket (not broadcast) — foundation for reconnection slice
- router dispatches game:ready to handleGameReady handler

packages/shared/src/schemas/lobby.ts

@@ -52,6 +52,12 @@ export const WsLobbyStartSchema = z.object({
 
 export type WsLobbyStart = z.infer<typeof WsLobbyStartSchema>;
 
+export const WsGameReadySchema = z.object({
+  type: z.literal("game:ready"),
+  lobbyId: z.uuid(),
+});
+export type WsGameReady = z.infer<typeof WsGameReadySchema>;
+
 export const WsGameAnswerSchema = z.object({
   type: z.literal("game:answer"),
   lobbyId: z.uuid(),
@@ -66,6 +72,7 @@ export const WsClientMessageSchema = z.discriminatedUnion("type", [
   WsLobbyLeaveSchema,
   WsLobbyStartSchema,
   WsGameAnswerSchema,
+  WsGameReadySchema,
 ]);
 export type WsClientMessage = z.infer<typeof WsClientMessageSchema>;