SDK execution error in Agent SDK 0.2.27+ causes crash before API calls

## Description

After the Agent SDK was bumped to 0.2.27 (commit `0ed5eea` on Jan 30, 2025), the claude-code-action crashes with an SDK execution error before any API calls are made.

## Environment

- **Action version:** `@v1` (currently points to Agent SDK 0.2.27+)
- **Last working version:** SHA `01e756b` (Agent SDK 0.2.25)
- **Runner:** `ubuntu-latest`
- **Workflow trigger:** `pull_request`

## Error Output

```
{
"type": "result",
"subtype": "success",
"is_error": true,
"duration_ms": 372,
"num_turns": 1,
"total_cost_usd": 0,
"permission_denials": []
}
SDK execution error: 14 | depsCount: ${Q},
15 | deps: ${$}}`};var Pj={keyword:"dependencies",type:"object",schemaType:"object",error:EB.error,code(X){...
...
error: Claude Code process exited with code 1
at $ (/home/runner/work/_actions/anthropics/claude-code-action/v1/base-action/node_modules/@anthropic-ai/claude-agent-sdk/sdk.mjs:19:7668)
at emit (node:events:98:22)
at #handleOnExit (node:child_process:520:14)
```

## Key Observations

1. **`total_cost_usd: 0`** - No API calls are made before the crash
2. **`is_error: true`** with `subtype: "success"` - Contradictory status
3. **Error appears to be in JSON schema validation** (ajv library) based on the minified code referencing `dependencies`, `schemaType`, etc.
4. **Crash happens during SDK initialization**, not during API communication

## Reproduction

The issue occurs with a minimal workflow configuration:

```yaml
- name: Run Claude Code Review
uses: anthropics/claude-code-action@v1
with:
anthropic_api_key: ${{ secrets.ANTHROPIC_API_KEY }}
github_token: ${{ secrets.GITHUB_TOKEN }}
model: claude-opus-4-5-20251101
prompt: "Review this PR"
```

We also tested:
- ❌ Removing `allowed_tools` - still crashes
- ❌ Using `model` input instead of `claude_args` - still crashes
- ✅ Pinning to SHA `01e756b` (SDK 0.2.25) - works correctly

## Workaround

Pin to the last working SHA:

```yaml
uses: anthropics/claude-code-action@01e756b34ef7a1447e9508f674143b07d20c2631
```

## Timeline

| Commit | Date | SDK Version | Status |
|--------|------|-------------|--------|
| `01e756b` | Jan 29 | 0.2.25 | ✅ Working |
| `0ed5eea` | Jan 30 | 0.2.27 | ❌ Broken |
| `70e16de` | Jan 31 | 0.2.29 | ❌ Broken |
| `4ce5f17` | Feb 1 | Latest | ❌ Broken |

## Expected Behavior

The action should successfully initialize and make API calls to perform code review.

## Actual Behavior

The action crashes with an SDK execution error during initialization, before any API calls are made.