feat(auto-mode): implement facade caching and enhance error handling

- Added caching for facades in AutoModeServiceCompat to persist auto loop state across API calls.
- Improved error handling in BoardView for starting and stopping auto mode, with user-friendly toast notifications.
- Updated WorktreePanel to manage auto mode state and concurrency limits more effectively.
- Enhanced useAutoMode hook to prevent state overwrites during transitions and synchronize UI with backend status.

This update optimizes performance and user experience in the auto mode feature.

apps/ui/src/components/views/board-view.tsx

@@ -1362,10 +1362,16 @@ export function BoardView() {
           if (enabled) {
             autoMode.start().catch((error) => {
               logger.error('[AutoMode] Failed to start:', error);
+              toast.error('Failed to start auto mode', {
+                description: error instanceof Error ? error.message : 'Unknown error',
+              });
             });
           } else {
             autoMode.stop().catch((error) => {
               logger.error('[AutoMode] Failed to stop:', error);
+              toast.error('Failed to stop auto mode', {
+                description: error instanceof Error ? error.message : 'Unknown error',
+              });
             });
           }
         }}