feat: 메인 화면에 정식 기수 표시

app/game-api/src/context.ts

@@ -44,6 +44,7 @@ export type WorldStateConfig = z.infer<typeof zWorldStateConfig>;
 
 export const zWorldStateMeta = z.object({
     serverId: z.string().optional(),
+    gameIdx: z.number().int().positive().optional(),
     starttime: z.string().optional(),
     opentime: z.string().optional(),
     preopenAt: z.string().optional(),

app/game-api/src/router/lobby/index.ts

@@ -53,6 +53,8 @@ export const lobbyRouter = router({
 
         return {
             serverId: worldState.meta.serverId?.trim() || ctx.profile?.name || 'game',
+            profile: ctx.profile.id,
+            gameIdx: worldState.meta.gameIdx ?? 1,
             year: worldState.currentYear,
             month: worldState.currentMonth,
             userCnt,