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>

test/permissions.test.ts

@@ -67,6 +67,7 @@ describe("checkWritePermissions", () => {
       branchPrefix: "claude/",
       useStickyComment: false,
       useCommitSigning: false,
+      botId: "41898282",
       allowedBots: "",
       trackProgress: false,
     },