feat(gateway): 프로필별 첫 기수 번호를 관리한다

첫 기수 번호를 완료 게임 수의 기준값으로 적용하고 0기를 API와 화면에서 보존한다. 시즌 번호와 독립된 RESET 계약을 관리 패널, 테스트, 운영 문서에 반영한다.

app/gateway-api/src/adminRouter.ts

@@ -1876,6 +1876,7 @@ export const adminRouter = router({
                         inGameNotice: z.string().max(4000).nullable().optional(),
                         profileImageUrl: z.string().max(2048).nullable().optional(),
                         nextSeasonIdx: z.number().int().min(0).nullable().optional(),
+                        firstGameIdx: z.number().int().min(0).nullable().optional(),
                         localAccountAccessGraceDays: z.number().int().min(0).max(365).nullable().optional(),
                         localAccountGeneralCreationGraceDays: z.number().int().min(0).max(365).nullable().optional(),
                         resetDefaults: zProfileResetDefaults.nullable().optional(),

app/gateway-api/src/orchestrator/gatewayOrchestrator.ts

@@ -271,6 +271,12 @@ const readMetaNumber = (meta: Record<string, unknown>, key: string): number | nu
     return null;
 };
 
+export const resolveProfileFirstGameIdx = (meta: Record<string, unknown>): number => {
+    const raw = meta.firstGameIdx;
+    const configured = typeof raw === 'number' ? raw : typeof raw === 'string' ? Number(raw) : Number.NaN;
+    return Number.isInteger(configured) && configured >= 0 ? configured : 1;
+};
+
 const normalizeStatus = (value: unknown): GatewayAdminActionStatus | null => {
     if (typeof value === 'string') {
         return value as GatewayAdminActionStatus;
@@ -1882,6 +1888,7 @@ export class GatewayOrchestrator implements GatewayOrchestratorHandle {
             );
             const profileMeta = normalizeMeta(profile.meta);
             const nextSeasonIdx = readMetaNumber(profileMeta, 'nextSeasonIdx');
+            const firstGameIdx = resolveProfileFirstGameIdx(profileMeta);
             const baseSeason = readMetaNumber(normalizeMeta(seedInfo.meta), 'season');
             const season = nextSeasonIdx ?? baseSeason ?? 1;
             await updateClaimedProfile({ status: 'STOPPED' }, () =>
@@ -1902,6 +1909,7 @@ export class GatewayOrchestrator implements GatewayOrchestratorHandle {
                     installOptions: {
                         ...(installOptions ?? {}),
                         season,
+                        firstGameIdx,
                         serverId,
                         installCommitSha: commitSha,
                     },