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>

test/mockContext.ts

@@ -9,6 +9,7 @@ import type {
   PullRequestReviewEvent,
   PullRequestReviewCommentEvent,
 } from "@octokit/webhooks-types";
+import { CLAUDE_APP_BOT_ID, CLAUDE_BOT_LOGIN } from "../src/github/constants";
 
 const defaultInputs = {
   prompt: "",
@@ -18,6 +19,8 @@ const defaultInputs = {
   branchPrefix: "claude/",
   useStickyComment: false,
   useCommitSigning: false,
+  botId: String(CLAUDE_APP_BOT_ID),
+  botName: CLAUDE_BOT_LOGIN,
   allowedBots: "",
   trackProgress: false,
 };