feat: integrate Claude Code SDK to replace process spawning (#327)

* feat: integrate Claude Code SDK to replace process spawning

- Add @anthropic-ai/claude-code dependency to base-action
- Replace mkfifo/cat process spawning with direct SDK usage
- Remove global Claude Code installation from action.yml files
- Maintain full compatibility with existing options
- Add comprehensive tests for SDK integration

This change makes the implementation cleaner and more reliable by
eliminating the complexity of managing child processes and named pipes.

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

Co-Authored-By: Claude <noreply@anthropic.com>

* fix: add debugging and bun executable for Claude Code SDK

- Add stderr handler to capture CLI errors
- Explicitly set bun as the executable for the SDK
- This should help diagnose why the CLI is exiting with code 1

* fix: extract mcpServers from parsed MCP config

The SDK expects just the servers object, not the wrapper object with mcpServers property.

* tsc

---------

Co-authored-by: Claude <noreply@anthropic.com>

base-action/action.yml

@@ -113,10 +113,6 @@ runs:
         cd ${GITHUB_ACTION_PATH}
         bun install
 
-    - name: Install Claude Code
-      shell: bash
-      run: npm install -g @anthropic-ai/claude-code@1.0.58
-
     - name: Run Claude Code Action
       shell: bash
       id: run_claude