Claude Code exits with code 1 after starting successfully - no actual work completed

## Description

Claude Code GitHub Action consistently exits with code 1 after starting successfully. Claude responds to @claude mentions and posts comments, but every run fails with "Claude Code process exited with code 1" after ~300-400ms of execution, regardless of task complexity.

## Environment

- **Action version:** `anthropics/claude-code-action@v1`
- **Claude Code version:** 2.0.74 (installed by action)
- **Runner:** `ubuntu-latest`
- **Authentication:** `anthropic_api_key` (API key from console.anthropic.com)
- **Repository type:** Yarn workspaces monorepo with Astro builds

## Workflow Configuration

Simplified workflow based on official example:

```yaml
name: Claude Code (Simple)

on:
issue_comment:
types: [created]
pull_request_review_comment:
types: [created]
issues:
types: [opened, assigned]
pull_request_review:
types: [submitted]

jobs:
claude:
if: |
(github.event_name == 'issue_comment' && contains(github.event.comment.body, '@claude')) ||
(github.event_name == 'pull_request_review_comment' && contains(github.event.comment.body, '@claude')) ||
(github.event_name == 'pull_request_review' && contains(github.event.review.body, '@claude')) ||
(github.event_name == 'issues' && (contains(github.event.issue.body, '@claude') || contains(github.event.issue.title, '@claude')))
runs-on: ubuntu-latest
permissions:
contents: write
pull-requests: write
issues: write
id-token: write
actions: read
steps:
- name: Checkout repository
uses: actions/checkout@v5
with:
fetch-depth: 0

- name: Run Claude Code
uses: anthropics/claude-code-action@v1
with:
anthropic_api_key: ${{ secrets.ANTHROPIC_API_KEY }}
```

## Expected Behavior

Claude should process the task and either:
- Complete the task successfully
- Provide a meaningful error message if it cannot complete the task
- Exit with code 0 on success

## Actual Behavior

Every single run:
1. ✅ Claude Code installs successfully
2. ✅ Authentication validates
3. ✅ Claude creates a branch (e.g., `claude/issue-48-20251228-2345`)
4. ✅ Claude posts a comment: "**Claude encountered an error** —— I'll analyze this and get back to you."
5. ❌ Exits with code 1 after ~300-400ms
6. ❌ Never completes any actual work

## Logs

Key excerpts from workflow run:

```
SDK options: {
"systemPrompt": {
"type": "preset",
"preset": "claude_code"
},
"pathToClaudeCodeExecutable": "",
"extraArgs": {
"auto-approve": null
},
"settingSources": [
"user",
"project",
"local"
]
}

[~2 seconds later]

SDK execution error: 13243 | throw error;
13244 | }
13245 | }
13246 | getProcessExitError(code, signal) {
13247 | if (code !== 0 && code !== null) {
13248 | return new Error(`Claude Code process exited with code ${code}`);
^
error: Claude Code process exited with code 1
at getProcessExitError (/home/runner/work/_actions/anthropics/claude-code-action/v1/base-action/node_modules/@anthropic-ai/claude-agent-sdk/sdk.mjs:13248:14)
at exitHandler (/home/runner/work/_actions/anthropics/claude-code-action/v1/base-action/node_modules/@anthropic-ai/claude-agent-sdk/sdk.mjs:13373:28)
```

Statistics from run:
```json
{
"duration_ms": 338,
"num_turns": 1,
"total_cost_usd": 0,
"permission_denials": []
}
```

**Note:** Even with `show_full_output: true` enabled, no actual Claude Code stderr/stdout is shown in the logs between "SDK options" and "SDK execution error".

## Steps to Reproduce

1. Set up workflow with configuration above
2. Add valid `ANTHROPIC_API_KEY` to repository secrets
3. Comment `@claude ` on an issue
4. Observe workflow run fails with exit code 1

Tested with various tasks:
- Simple questions: "@claude please explain what this issue is asking for"
- File creation: "@claude please create a simple README.md file"
- Complex tasks: Original issue requesting domain linking analysis

All fail identically.

## What We've Tried

### Authentication Methods
- ❌ `claude_code_oauth_token` parameter (validation error)
- ❌ `ANTHROPIC_API_KEY` env var at job level (validation error)
- ❌ `CLAUDE_CODE_OAUTH_TOKEN` env var at step level (empty)
- ❌ `CLAUDE_CODE_OAUTH_TOKEN` env var at job level (empty)
- ✅ `anthropic_api_key` input parameter (works, but exits with code 1)

### Configuration Changes
- Switched from custom OAuth token to API key
- Simplified workflow to match official example exactly
- Removed all custom Node.js/yarn setup steps
- Regenerated API credentials
- Tested on fresh issues

### Debugging
- Enabled `show_full_output: true` - no additional output shown
- Checked for API authentication errors - none found
- Tested with simple and complex tasks - all fail identically

## Additional Context

The action is technically "working" in that:
- Authentication succeeds
- Claude responds to mentions
- Branches are created
- Comments are posted

But it never completes any actual work due to the immediate exit code 1.

The $0 cost in the statistics suggests Claude might not be making API calls successfully, or the run is too short to register cost.

## Questions

1. Is there a way to see the actual Claude Code CLI stdout/stderr even with `show_full_output: true`?
2. Are there any known issues with exit code 1 in GitHub Actions environments?
3. Should the action be logging more details about why Claude Code exited?

## Related Issues

Similar "exit code 1" errors reported in other contexts:
- #522 (App token exchange failed)
- #722 (First-time setup workflow failures)

However, those are different scenarios - our issue occurs on every run, not just first-time setup.