feat: integrate Discord bot and Dispatcher in FastAPI lifespan

Adds asynccontextmanager lifespan to webapp.py that:
- Initializes task_queue and message_store on startup
- Starts Dispatcher as background task
- Starts Discord bot gateway if DISCORD_TOKEN is set
- Properly shuts down all resources on application exit

This completes Phase 3 Task 6, enabling the webapp to run Discord bot
and dispatcher concurrently with the webhook server.

agent/webapp.py

@@ -1,10 +1,12 @@
 """galaxis-agent webhook server."""
+import asyncio
 import hashlib
 import hmac
 import json
 import logging
 import os
 import re
+from contextlib import asynccontextmanager
 
 from fastapi import FastAPI, Request, HTTPException
 from slowapi import Limiter
@@ -14,7 +16,41 @@ from slowapi.middleware import SlowAPIMiddleware
 
 logger = logging.getLogger(__name__)
 
-app = FastAPI(title="galaxis-agent")
+
+@asynccontextmanager
+async def lifespan(app: FastAPI):
+    """애플리케이션 시작/종료 시 리소스를 관리한다."""
+    from agent.task_queue import get_task_queue
+    from agent.message_store import get_message_store
+    from agent.dispatcher import Dispatcher
+    from agent.integrations.discord_handler import DiscordHandler
+
+    task_queue = await get_task_queue()
+    message_store = await get_message_store()
+
+    dispatcher = Dispatcher(task_queue=task_queue)
+    await dispatcher.start()
+    app.state.dispatcher = dispatcher
+
+    discord_token = os.environ.get("DISCORD_TOKEN", "")
+    discord_handler = None
+    if discord_token:
+        discord_handler = DiscordHandler()
+        discord_task = asyncio.create_task(discord_handler.start(discord_token))
+        app.state.discord_handler = discord_handler
+        logger.info("Discord bot starting...")
+
+    yield
+
+    await dispatcher.stop()
+    if discord_handler:
+        await discord_handler.close()
+    await task_queue.close()
+    await message_store.close()
+    logger.info("Application shutdown complete")
+
+
+app = FastAPI(title="galaxis-agent", lifespan=lifespan)
 limiter = Limiter(key_func=get_remote_address)
 app.state.limiter = limiter
 app.add_middleware(SlowAPIMiddleware)