[server] Board API Logic

> ### ISSUE
> * Group: `server`
> * Type: `feature`
> * Detail: board API method logic 작성
>
> ### TODO
> 1. [ ] context(GET)
> 2. [ ] writeContext(POST)
> 3. [ ] deleteContext(DELETE)
>
> ### Estimated time
>
> ### `0.5h`
>
> ### Labels
> * Estimated time: `E: 0.5h`
> * Group : `server`
> * Sprint: `Sprint__2__`
> * Urgency: `Middle`

Start by locating the server-side board API entry points corresponding to context (GET), writeContext (POST), and deleteContext (DELETE). Implement the three listed API methods and verify that each endpoint behaves as intended; the issue names no files or tests, so the project’s existing server API conventions must be identified first.

Written by the indexing model from the issue text.