Migrate the Claude common config snippet storage from browser localStorage
to the persistent config.json file for better cross-device sync and backup support.
**Backend Changes:**
- Add `claude_common_config_snippet` field to `MultiAppConfig` struct
- Add `get_claude_common_config_snippet` and `set_claude_common_config_snippet` Tauri commands
- Include JSON validation in the setter command
**Frontend Changes:**
- Create new `lib/api/config.ts` API module
- Refactor `useCommonConfigSnippet` hook to use config.json instead of localStorage
- Add automatic one-time migration from localStorage to config.json
- Add loading state during initialization
**Benefits:**
- Cross-device synchronization via backup/restore
- More reliable persistence than browser storage
- Centralized configuration management
- Seamless migration for existing users
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