[Task] LandingPage / 로그인, 로그아웃, 회원가입

### ISSUE
- Group: `client`
- Type: `feature`
- Detail: 로그인, 로그아웃, 회원가입 기능 구현

### TODO
1. [ ] 회원가입 기능 구현
2. [ ] 로그인 기능 구현
3. [ ] 로그아웃 기능 구현

### Estimated time

`1h`

### Labels
- Estimated time: `E: 1h`
- Group : `client`
- Sprint: `Sprint__NUMBER__`
- Urgency: `High`

Start at the client LandingPage entry point and inspect how authentication-related UI and state are currently organized. Implement the three listed flows—sign-up, login, and logout—and verify that each works from the LandingPage.

Written by the indexing model from the issue text.