feat: update doc-check workflow to utilize claude-skills (#21588)

## Summary

Updates our existing doc-check workflow to utilize new claude-skills
that exist in the repository for better contextual behavior with less
prompting requirements in the workflow

## Changes

### New: Claude Skill (`.claude/skills/doc-check/SKILL.md`)

Defines the doc-check workflow for Claude:
- Check PR title first to skip irrelevant reviews (refactors, tests,
chores)
- Read code changes and search existing docs for related content
- Post structured comments with specific recommendations

### Updated: GitHub Workflow (`.github/workflows/doc-check.yaml`)

- **Triggers**: PR opened, updated (synchronize), `doc-check` label
added, or manual dispatch
- **Task lifecycle**: Creates task → monitors completion → fetches logs
→ cleans up
- **Context-aware prompts**: Tells Claude if this is a new PR, update,
or manual request
- Uses `coder-username` parameter to run as `doc-check-bot` service
account

.claude/skills/doc-check/SKILL.md

@@ -0,0 +1,79 @@
+---
+name: doc-check
+description: Checks if code changes require documentation updates
+---
+
+# Documentation Check Skill
+
+Review code changes and determine if documentation updates or new documentation
+is needed.
+
+## Workflow
+
+1. **Get the code changes** - Use the method provided in the prompt, or if none
+   specified:
+   - For a PR: `gh pr diff <PR_NUMBER> --repo coder/coder`
+   - For local changes: `git diff main` or `git diff --staged`
+   - For a branch: `git diff main...<branch>`
+
+2. **Understand the scope** - Consider what changed:
+   - Is this user-facing or internal?
+   - Does it change behavior, APIs, CLI flags, or configuration?
+   - Even for "internal" or "chore" changes, always verify the actual diff
+
+3. **Search the docs** for related content in `docs/`
+
+4. **Decide what's needed**:
+   - Do existing docs need updates to match the code?
+   - Is new documentation needed for undocumented features?
+   - Or is everything already covered?
+
+5. **Report findings** - Use the method provided in the prompt, or if none
+   specified, summarize findings directly
+
+## What to Check
+
+- **Accuracy**: Does documentation match current code behavior?
+- **Completeness**: Are new features/options documented?
+- **Examples**: Do code examples still work?
+- **CLI/API changes**: Are new flags, endpoints, or options documented?
+- **Configuration**: Are new environment variables or settings documented?
+- **Breaking changes**: Are migration steps documented if needed?
+- **Premium features**: Should docs indicate `(Premium)` in the title?
+
+## Key Documentation Info
+
+- **`docs/manifest.json`** - Navigation structure; new pages MUST be added here
+- **`docs/reference/cli/*.md`** - Auto-generated from Go code, don't edit directly
+- **Premium features** - H1 title should include `(Premium)` suffix
+
+## Coder-Specific Patterns
+
+### Callouts
+
+Use GitHub-Flavored Markdown alerts:
+
+```markdown
+> [!NOTE]
+> Additional helpful information.
+
+> [!WARNING]
+> Important warning about potential issues.
+
+> [!TIP]
+> Helpful tip for users.
+```
+
+### CLI Documentation
+
+CLI docs in `docs/reference/cli/` are auto-generated. Don't suggest editing them
+directly. Instead, changes should be made in the Go code that defines the CLI
+commands (typically in `cli/` directory).
+
+### Code Examples
+
+Use `sh` for shell commands:
+
+```sh
+coder server --flag-name value
+```