[FEATURE] Add "Execute on Web" option when exiting Plan Mode

Preflight Checklist

• [x] I have searched existing requests and this feature hasn't been requested yet
• [x] This is a single feature request (not multiple features)

Problem Statement

After creating a plan in Plan Mode, users currently have to manually type & Execute the plan
to send execution to Claude Code on the web. This breaks the flow and requires remembering the syntax.

Proposed Solution

Add a new option when exiting Plan Mode:

Plan completed. How would you like to proceed?

1. Clear context and Execute locally(auto-accept)
2. Execute locally (auto-accept)
3. Execute locally (manual approval)
4. Execute on web (background task) ← NEW
5. No, keep planning

<img width="463" height="127" alt="Image" src="https://github.com/user-attachments/assets/191b4a80-765d-48d8-9fdc-53213975c5ee" />

Alternative Solutions

_No response_

Priority

Medium - Would be very helpful

Feature Category

Interactive mode (TUI)

Use Case Example

• Seamless "plan locally, execute remotely" workflow
• No need to remember & prefix syntax
• Consistent with the official best practice pattern documented at

https://code.claude.com/docs/en/claude-code-on-the-web

Additional Context

_No response_