Background Agents Cannot Access Skill Tool

## Environment

- **Claude Agent SDK Version:** 0.1.67
- **Claude Code Version:** 2.0.67

## Summary

Background/asynchronous agents spawned with `run_in_background: true` do not have access to the `Skill` tool, even when explicitly specified in the agent definition.

## Description

When using the Claude Agent SDK to spawn agents programmatically, there is an inconsistency in tool availability between synchronous and asynchronous agents:

- **Synchronous agents**: Have access to the `Skill` tool as expected
- **Asynchronous agents** (background): Do NOT have access to the `Skill` tool, despite having access to other tools like `Read`, `Write`, and `Bash`

This prevents background agents from loading and executing skills, limiting their functionality compared to synchronous agents.

### Code Example

```typescript
const messages = query({
prompt: "Use the my-agent agent in the background to write a fun fact about cats.",
options: {
settingSources: ["project"],
plugins: [{ type: "local", path: "./plugins" }],
},
});
```

## Attempted Workarounds

The following approaches were tested without success:

1. **Frontmatter specification**: Listing the `Skill` tool in the agent definition markdown file frontmatter
```yaml
---
name: my-agent
tools: Skill,Read,Write,Bash
---
```

2. **Agent definition object**: Setting tools in the `AgentDefinition` object passed to the query function
```typescript
definition.tools = ["Skill", "Read", "Write", "Bash"];
agents["my-agent"] = definition;
```

None of these approaches successfully granted the `Skill` tool access to background agents.

## Expected Behavior

Background agents should have access to the `Skill` tool when it is:
- Explicitly listed in the agent definition's `tools` property
- Specified in the agent markdown file frontmatter

The `Skill` tool should be available to background agents in the same way it is available to synchronous agents, allowing them to load and execute skills as needed.

## Actual Behavior

Background agents do not have access to the `Skill` tool, regardless of how it is specified in the configuration. Other tools (e.g., `Read`, `Write`, `Bash`) are accessible to background agents as expected.

## Additional Context

The issue appears to be specific to the `Skill` tool and does not affect other tools, suggesting there may be special handling of the `Skill` tool that doesn't account for background agent execution contexts.

Available tools were determined by inspecting the requests to the LiteLLM proxy.