feat: add bot_id input to handle GitHub App authentication errors (#534)

Adds a new optional bot_id input parameter that defaults to the github-actions[bot] ID (41898282). This resolves the "403 Resource not accessible by integration" error that occurs when using GitHub App installation tokens, which cannot access the /user endpoint.

Changes:
- Add bot_id input to action.yml with default value
- Update context parsing to include bot_id from environment
- Modify agent mode to use bot_id when available, avoiding API calls that fail with GitHub App tokens
- Add clear error handling for GitHub App token limitations
- Update documentation in usage.md and faq.md
- Fix test mocks to include bot_id field

This allows users to specify a custom bot user ID or use the default github-actions[bot] ID automatically, preventing 403 errors in automation workflows.

🤖 Generated with [Claude Code](https://claude.ai/code)

Co-authored-by: github-actions[bot] <41898282+github-actions[bot]@users.noreply.github.com>
Co-authored-by: Claude <noreply@anthropic.com>

src/modes/agent/index.ts

@@ -77,22 +77,16 @@ export const agentMode: Mode = {
     return false;
   },
 
-  async prepare({
-    context,
-    githubToken,
-    octokit,
-  }: ModeOptions): Promise<ModeResult> {
+  async prepare({ context, githubToken }: ModeOptions): Promise<ModeResult> {
     // Configure git authentication for agent mode (same as tag mode)
     if (!context.inputs.useCommitSigning) {
-      try {
-        // Get the authenticated user (will be claude[bot] when using Claude App token)
-        const { data: authenticatedUser } =
-          await octokit.rest.users.getAuthenticated();
-        const user = {
-          login: authenticatedUser.login,
-          id: authenticatedUser.id,
-        };
+      // Use bot_id and bot_name from inputs directly
+      const user = {
+        login: context.inputs.botName,
+        id: parseInt(context.inputs.botId),
+      };
 
+      try {
         // Use the shared git configuration function
         await configureGitAuth(githubToken, context, user);
       } catch (error) {