feat: Add debug mode to inspect LLM prompts and responses

- Add DebugContext to track all LLM interactions
- Create DebugPanel component with floating UI
- Track prompts/responses in Tutor component (move analysis, hints, questions)
- Track prompts/responses in Analysis page (move commentary)
- Add NEXT_PUBLIC_DEBUG environment variable to enable/disable
- Include metadata (FEN, personality, language, etc.) in debug entries
- Add copy-to-clipboard and clear functionality
- Add DEBUG_MODE.md documentation

DEBUG_MODE.md

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+# Debug Mode
+
+Debug mode allows you to inspect the exact prompts sent to the LLM and the responses received. This is useful for troubleshooting AI behavior and understanding how the tutor works.
+
+## Enabling Debug Mode
+
+1. Create a `.env.local` file in the root directory (if it doesn't exist)
+2. Add the following line:
+   ```
+   NEXT_PUBLIC_DEBUG=true
+   ```
+3. Restart the development server (`npm run dev`)
+
+## Using Debug Mode
+
+When debug mode is enabled, you'll see:
+
+### Floating Debug Panel
+- A purple "Debug Mode" button appears in the bottom-right corner
+- Click to expand and see all LLM interactions
+- Shows a list of all prompts sent during your session
+- Click any entry to see the full prompt and response
+
+### Features
+- **Copy to Clipboard**: Copy prompts or responses for analysis
+- **Clear Entries**: Clear all debug entries
+- **Metadata**: View additional context like FEN, personality, language, etc.
+- **Timestamps**: See when each interaction occurred
+
+### What's Tracked
+
+**Tutor Component:**
+- Move Analysis (automatic after each move)
+- Best Move Requests (when user asks for the best move)
+- Hint Requests (when user asks for a hint)
+- General Questions (any other user question)
+
+**Analysis Page:**
+- Move Analysis for each move in the game
+- Includes move number, color, FEN before/after, evaluation changes
+
+## Disabling Debug Mode
+
+1. Remove or comment out the `NEXT_PUBLIC_DEBUG=true` line in `.env.local`
+2. Or set it to `false`: `NEXT_PUBLIC_DEBUG=false`
+3. Restart the development server
+
+## Privacy Note
+
+Debug mode only runs locally in your browser. No debug data is sent to any server.
+