feat: implement async SQLAlchemy database engine and session management with Neon-compatible configurations

backend/src/database.py

@@ -39,6 +39,9 @@ engine = create_async_engine(
     _db_url,
     echo=False,
     pool_pre_ping=True,
+    pool_recycle=300,        # Drop idle connections aggressively (every 5 mins) to prevent Neon Serverless drops
+    pool_size=10,            # Constrain pool size for Free Tier
+    max_overflow=5,
     connect_args=_connect_args,
     # This disables SQLAlchemy's internal prepared statement cache globally
     execution_options={"prepared_statement_cache_size": 0}