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/install-mcp-server.test.ts

@@ -2,6 +2,7 @@ import { describe, test, expect, beforeEach, afterEach, spyOn } from "bun:test";
 import { prepareMcpConfig } from "../src/mcp/install-mcp-server";
 import * as core from "@actions/core";
 import type { ParsedGitHubContext } from "../src/github/context";
+import { CLAUDE_APP_BOT_ID, CLAUDE_BOT_LOGIN } from "../src/github/constants";
 
 describe("prepareMcpConfig", () => {
   let consoleInfoSpy: any;
@@ -31,6 +32,8 @@ describe("prepareMcpConfig", () => {
       branchPrefix: "",
       useStickyComment: false,
       useCommitSigning: false,
+      botId: String(CLAUDE_APP_BOT_ID),
+      botName: CLAUDE_BOT_LOGIN,
       allowedBots: "",
       trackProgress: false,
     },