fix(multi): sync game mode changes to server in lobby

app/api/rooms/[code]/route.ts

@@ -127,7 +127,7 @@ export async function PATCH(
     article?: string;
     startArticle?: string;
     targetArticle?: string;
-    value?: boolean;
+    value?: boolean | number | string;
   };
 
   // Nettoyer les joueurs inactifs uniquement hors partie (lobby/résultats)
@@ -351,6 +351,19 @@ export async function PATCH(
       return Response.json({ room });
     }
 
+    // Hôte change le mode de jeu
+    case "setGameMode": {
+      const host = room.players.find((p) => p.id === playerId);
+      if (!host?.isHost) {
+        return Response.json({ error: "Seul l'hôte peut modifier ce paramètre" }, { status: 403 });
+      }
+      if (value === "race" || value === "all_finish") {
+        room.gameMode = value;
+        await saveRoom(room);
+      }
+      return Response.json({ room });
+    }
+
     // Hôte change la limite de temps
     case "setTimeLimit": {
       const host = room.players.find((p) => p.id === playerId);

app/components/LobbyScreen.tsx

@@ -18,6 +18,7 @@ type LobbyScreenProps = {
   setTotalRounds: (v: number) => void;
   gameMode: Room["gameMode"];
   setGameMode: (v: Room["gameMode"]) => void;
+  onSetGameMode: (v: Room["gameMode"]) => void;
   onSetSearchAllowed: (v: boolean) => void;
   onSetTimeLimit: (v: number) => void;
 };
@@ -37,6 +38,7 @@ export function LobbyScreen({
   setTotalRounds,
   gameMode,
   setGameMode,
+  onSetGameMode,
   onSetSearchAllowed,
   onSetTimeLimit,
 }: LobbyScreenProps) {
@@ -151,7 +153,7 @@ export function LobbyScreen({
             <span className="text-xs text-[#888]">Mode de jeu</span>
             <select
               value={gameMode}
-              onChange={(e) => setGameMode(e.target.value as Room["gameMode"])}
+              onChange={(e) => { const v = e.target.value as Room["gameMode"]; setGameMode(v); onSetGameMode(v); }}
               className="w-full min-h-11 px-2 bg-[#0f0f0f] border border-[#2e2e2e] rounded-xl text-sm text-[#f0f0f0] outline-none focus:border-[#7c3aed] cursor-pointer transition-colors"
             >
               <option value="race">Course - le premier arrivé gagne</option>

app/components/ScreenRouter.tsx

@@ -158,6 +158,7 @@ export function ScreenRouter({
         setTotalRounds={setTotalRounds}
         gameMode={gameMode}
         setGameMode={setGameMode}
+        onSetGameMode={(v) => multi.setGameMode(v)}
         onSetSearchAllowed={(v) => multi.setSearchAllowed(v)}
         onSetTimeLimit={(v) => multi.setTimeLimit(v)}
       />