feat: add bot_id input to handle GitHub App authentication errors

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: Claude <noreply@anthropic.com>

action.yml

@@ -73,6 +73,10 @@ inputs:
     description: "Enable commit signing using GitHub's commit signature verification. When false, Claude uses standard git commands"
     required: false
     default: "false"
+  bot_id:
+    description: "GitHub user ID to use for git operations when authenticated user cannot be fetched (defaults to github-actions[bot] ID)"
+    required: false
+    default: "41898282"
   track_progress:
     description: "Force tag mode with tracking comments for pull_request and issue events. Only applicable to pull_request (opened, synchronize, ready_for_review, reopened) and issue (opened, edited, labeled, assigned) events."
     required: false
@@ -144,6 +148,7 @@ runs:
         USE_STICKY_COMMENT: ${{ inputs.use_sticky_comment }}
         DEFAULT_WORKFLOW_TOKEN: ${{ github.token }}
         USE_COMMIT_SIGNING: ${{ inputs.use_commit_signing }}
+        BOT_ID: ${{ inputs.bot_id }}
         TRACK_PROGRESS: ${{ inputs.track_progress }}
         ADDITIONAL_PERMISSIONS: ${{ inputs.additional_permissions }}
         CLAUDE_ARGS: ${{ inputs.claude_args }}