fix(tui): hide background hint when all work is already backgrounded

[aakankshaduggal]

Issue for this PR

Fixes #36940

Type of change

• Bug fix
• New feature
• Refactor / code improvement
• Documentation

What does this PR do?

The ctrl+b background hint checked state.metadata.background but metadata is set asynchronously via ctx.metadata(). There's a window where metadata is undefined even for tasks launched with background: true. This adds a check on state.input.background which is available immediately from the tool call parameters.

Extracts isForegroundRunningTask to deduplicate the filter in foregroundTasks memo and the inline <Show> condition.

How did you verify your code works?

• bun test --cwd packages/tui — 197 pass, 1 skip, 0 fail
• bun run --cwd packages/tui typecheck — clean
• bun turbo typecheck — 30/30 packages pass (ran via pre-push hook)
• Added 6 targeted tests covering: foreground task, background via input, background via metadata, both set, non-task tools, completed task

Checklist

• I have tested my changes locally
• I have not included unrelated changes in this PR

[github-actions]

The following comment was made by an LLM, it may be inaccurate:

Potential Duplicate Found:

• PR fix(tui): hide background hint for async work #37088: fix(tui): hide background hint for async work
  • fix(tui): hide background hint for async work #37088

This appears to be a related or potentially duplicate fix addressing the same background hint visibility issue in the TUI. You should review PR #37088 to determine if it already addresses the issue or if your PR #39242 provides an improved/alternative solution.

[aakankshaduggal]

Note: PR #37088 addresses the same issue on the v2 branch. This PR targets dev (the default branch), which has different TUI code — dev uses an inline <Show> condition in AssistantMessage and a foregroundTasks memo, while v2 has a separate BackgroundToolHint component with toolDisplay() checks. The two PRs don't overlap.