Improve client matchmaking flow with ticket handling, auto-join, and mode distribution

### PlayerWebSocketHandler updates
- Track `ticket` and `match_id` per player instance
- Handle `matchmaker_ticket` messages and store ticket
- Handle `matchmaker_matched` and automatically join created match
- Enhance matchmaking debug output
- Update join_matchmaking() to include mode-based string_properties + query

### Matchmaking simulation improvements
- Evenly distribute players between "classic" and "blitz" modes
- Randomize assignment order to simulate real queue behavior
- Log player→mode mapping for visibility during tests

Example:
  player_0 -> classic
  player_3 -> blitz
  player_5 -> classic

Client test harness now accurately reflects multi-mode matchmaking behavior.

game_flow.py

@@ -96,7 +96,23 @@ class PlayerWebSocketHandler(WebSocketHandler):
         await handler.connect()
         return handler
 
+    def __init__(self, custom_id: str, label: str):
+        super().__init__(custom_id, label)
+        self.ticket = None
+        self.match_id = None
+
     async def on_message(self, msg):
+        if "matchmaker_matched" in msg:
+            match_id = msg["matchmaker_matched"]["match_id"]
+            print(f"[{self.label}] ✅ Match found: {match_id}")
+            await self.join_match(match_id)
+            return
+
+        if "matchmaker_ticket" in msg:
+            self.ticket = msg["matchmaker_ticket"]["ticket"]
+            print(f"[{self.label}] ✅ Received ticket: {self.ticket}")
+            return
+
         if "match_data" not in msg:
             print(f"[{self.label}] {msg}")
             return
@@ -124,9 +140,8 @@ class PlayerWebSocketHandler(WebSocketHandler):
             "matchmaker_add": {
                 "min_count": 2,
                 "max_count": 2,
-                "string_properties": {
-                    "mode": mode
-                }
+                "string_properties": {"mode": mode},
+                "query": f"+mode:{mode}"
             }
         }))
         print(f"[{self.label}] Searching match for mode={mode}...")
@@ -151,6 +166,7 @@ class PlayerWebSocketHandler(WebSocketHandler):
     async def join_match(self, match_id: str):
         await self.ws.send(json.dumps({"match_join": {"match_id": match_id}}))
         print(f"[{self.label}] Joined match: {match_id}")
+        self.match_id = match_id
 
     # ---------- Gameplay ----------
     async def send_move(self, match_id: str, row: int, col: int):