feat: Expose GitHub token as action output for external use

This allows workflows to use the Claude App token obtained by the action
for posting comments as claude[bot] instead of github-actions[bot].

Changes:
- Add github_token output to action.yml
- Export token from prepare.ts after authentication
- Allows workflows to use the same token Claude uses internally

action.yml

@@ -93,6 +93,9 @@ outputs:
   branch_name:
     description: "The branch created by Claude Code for this execution"
     value: ${{ steps.prepare.outputs.CLAUDE_BRANCH }}
+  github_token:
+    description: "The GitHub token used by the action (Claude App token if available)"
+    value: ${{ steps.prepare.outputs.github_token }}
 
 runs:
   using: "composite"

src/entrypoints/prepare.ts

@@ -62,6 +62,9 @@ async function run() {
 
     // MCP config is handled by individual modes (tag/agent) and included in their claude_args output
     
+    // Expose the GitHub token (Claude App token) as an output
+    core.setOutput("github_token", githubToken);
+
     // Step 6: Get system prompt from mode if available
     if (mode.getSystemPrompt) {
       const modeContext = mode.prepareContext(context, {