Feature Request: Bidirectional sync between Copilot CLI and VS Code Chat for shared sessions

### Describe the feature or problem you'd like to solve

Currently, when /ide is used to connect Copilot CLI to VS Code, messages typed in the CLI appear in VS Code chat, but messages sent from VS Code chat do not appear in the CLI console. This makes it impossible to use both interfaces together in the same session.

### Proposed solution

Both CLI and VS Code chat should act as equal input/output surfaces for the same session — messages from either side should be visible or processed from both.

Suggested Approach - Route VS Code chat inputs through the active `/ide`-connected CLI session so both
surfaces share a single message queue.

### Example prompts or workflows

Power users want to combine CLI capabilities (shell execution, /fleet, file tools, terminal access) with VS Code chat's inline editor integration (code lens, diff view, editor context) in the same working session without losing context.

### Additional context

_No response_

Start by tracing the /ide connection path and the VS Code chat integration; the issue does not name specific files or tests. Determine how the active CLI session currently receives and emits messages, then identify the shared-session and queue boundaries. Done means messages sent from either CLI or VS Code chat are visible and processed by both surfaces without losing session context.

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