Feature request: plan-approval parity on remote-control surfaces — offer "accept, clear context and auto mode"

## Problem

In the CLI, when a plan is ready for approval, the primary option is
**"accept, clear context and start in auto mode"** (especially with
`showClearContextOnPlanAccept: true`). This is extremely valuable: all the
exploration, file reading and brainstorming that produced the plan is
dropped, and implementation starts with a clean context that contains only
the plan itself.

When the same Claude Code session is controlled remotely — from the mobile
app, the desktop app, or claude.ai remote control — the plan-approval
prompt only offers a plain accept. The clear-context + auto-mode action is
not available.

The practical consequence: plans approved from a phone or from claude.ai
carry the entire research context into the implementation phase, burning
context window on content the plan has already distilled. The user has to
either walk back to the terminal to approve, or accept a degraded
implementation context.

This mirrors a broader pattern in plan-driven workflows: context is
cleared whenever state has been externalized (an approved plan,
consolidated docs). The plan-accept action is the one place the product
supports this pattern natively — and remote surfaces are missing exactly
that.

## Proposal

Expose the full set of plan-approval actions on every surface that can
control a session — at minimum the "accept, clear context and start in
auto mode" option, honoring the same `showClearContextOnPlanAccept`
setting the CLI uses.

Remote control is most useful precisely for long-running, plan-driven
work sessions; that is exactly the scenario where clearing the planning
context matters most.

Compare the CLI plan-approval flow, especially showClearContextOnPlanAccept, with plan approval in the mobile app, desktop app, and claude.ai remote control. Start by locating the shared approval entry points and the remote-surface actions. Done means every surface exposes the clear-context and auto-mode action while honoring the setting.

Written by the indexing model from the issue text.