feat: add multi-provider LLM support with LiteLLM

- Replace openai with litellm for unified LLM interface
- Support 100+ providers: OpenAI, Anthropic, Gemini, Azure, Ollama, etc.
- Add custom API base URL support (LLM_API_BASE)
- Add .env file support with python-dotenv
- Add .env.example configuration template

🤖 Generated with [Claude Code](https://claude.com/claude-code)

Co-Authored-By: Claude Opus 4.5 <noreply@anthropic.com>

backend/app/main.py

@@ -3,6 +3,10 @@ FastAPI entry point for the interactive live-stream game backend.
 Configures the application, WebSocket routes, and lifecycle events.
 """
 
+# Load .env file before any other imports
+from dotenv import load_dotenv
+load_dotenv()
+
 import logging
 from contextlib import asynccontextmanager
 from pathlib import Path