Add health checks and reconnection logic for stale daemon sockets (bd-137)

- Add ping() and health() methods to BdDaemonClient for connection verification
- Implement _health_check_client() to verify cached client connections
- Add _reconnect_client() with exponential backoff (0.1s, 0.2s, 0.4s, max 3 retries)
- Update _get_client() to health-check before returning cached clients
- Automatically detect and remove stale connections from pool
- Add comprehensive test suite with 14 tests covering all scenarios
- Handle daemon restarts, upgrades, and long-idle connections gracefully

Amp-Thread-ID: https://ampcode.com/threads/T-2366ef1b-389c-4293-8145-7613037c9dfa
Co-authored-by: Amp <amp@ampcode.com>

integrations/beads-mcp/src/beads_mcp/bd_daemon_client.py

@@ -200,10 +200,33 @@ class BdDaemonClient(BdClientBase):
 
         Raises:
             DaemonNotRunningError: If daemon is not running
+            DaemonConnectionError: If connection fails
+            DaemonError: If request fails
         """
         data = await self._send_request("ping", {})
         return json.loads(data) if isinstance(data, str) else data
 
+    async def health(self) -> Dict[str, Any]:
+        """Get daemon health status.
+
+        Returns:
+            Dict with health info including:
+            - status: "healthy" | "degraded" | "unhealthy"
+            - version: daemon version string
+            - uptime: uptime in seconds
+            - cache_size: number of cached databases
+            - db_response_time_ms: database ping time
+            - active_connections: number of active connections
+            - memory_bytes: memory usage
+
+        Raises:
+            DaemonNotRunningError: If daemon is not running
+            DaemonConnectionError: If connection fails
+            DaemonError: If request fails
+        """
+        data = await self._send_request("health", {})
+        return json.loads(data) if isinstance(data, str) else data
+
     async def init(self, params: Optional[InitParams] = None) -> str:
         """Initialize new beads database (not typically used via daemon).