### Context & Goal

The `exam` command currently contains a lot of business logic, including reward calculations, timer management, and complex database transactions. Moving this logic to a dedicated `ExamService` will improve testability, maintainability, and keep the command file focused on user interaction.

### Dependencies

- None

### Affected Files

- `shared/modules/economy/exam.service.ts`: New file for the exam logic.
- `bot/commands/economy/exam.ts`: Refactor to use the new service.

### Technical Constraints & Strategy

- Implementation: Create an `ExamService` that handles `getExamStatus`, `takeExam`, and `registerForExam`.
- The command should only handle user input and formatting the response embeds based on the service's result.
- Ensure the Drizzle transactions are correctly handled within the service.

### Definition of Done (Binary)

- [ ] `ExamService` implemented with methods for all exam-related operations.
- [ ] `bot/commands/economy/exam.ts` refactored to use the service.
- [ ] Logic is covered by unit tests in a new test file.
- [ ] Manual verification shows the exam command still works as expected.

### New Test Files

- `shared/modules/economy/exam.service.test.ts`: Unit tests for reward calculations and state transitions.