feat: add CalDAV synchronization with automatic sync

- Add CaldavService with tsdav/ical.js for CalDAV server communication
- Add CaldavController, CaldavRepository, and caldav routes
- Add client-side CaldavConfigService with sync(), config CRUD
- Add CalDAV settings UI with config load/save in settings screen
- Sync on login, auto-login (AuthGuard), periodic timer (calendar), and sync button
- Push single events to CalDAV on server-side create/update/delete
- Push all events to CalDAV after chat event confirmation
- Refactor ChatService to use EventService instead of direct EventRepository
- Rename CalDav/calDav to Caldav/caldav for consistent naming
- Add Radicale Docker setup for local CalDAV testing
- Update PlantUML diagrams and CLAUDE.md with CalDAV architecture

packages/shared/src/utils/formatters.ts

@@ -69,3 +69,11 @@ export function formatDateWithWeekdayShort(date: Date): string {
     month: "2-digit",
   });
 }
+
+// Format date as YYYY-MM-DD for exception date comparison
+export function formatDateKey(date: Date): string {
+  const year = date.getFullYear();
+  const month = String(date.getMonth() + 1).padStart(2, "0");
+  const day = String(date.getDate()).padStart(2, "0");
+  return `${year}-${month}-${day}`;
+}