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

apps/server/src/ai/utils/eventFormatter.ts

@@ -1,4 +0,0 @@
-import { formatDate, formatTime, formatDateTime } from "@calchat/shared";
-
-// Re-export from shared package for use in toolExecutor
-export { formatDate, formatTime, formatDateTime };

apps/server/src/ai/utils/index.ts

@@ -1,4 +1,3 @@
-export { formatDate, formatTime, formatDateTime } from "./eventFormatter";
 export { buildSystemPrompt } from "./systemPrompt";
 export {
   TOOL_DEFINITIONS,

apps/server/src/ai/utils/toolExecutor.ts

@@ -6,7 +6,7 @@ import {
   RecurringDeleteMode,
 } from "@calchat/shared";
 import { AIContext } from "../../services/interfaces";
-import { formatDate, formatTime, formatDateTime } from "./eventFormatter";
+import { formatDate, formatTime, formatDateTime } from "@calchat/shared";
 
 /**
  * Check if two time ranges overlap.