Add improve and verify stages to the task pipeline

[lee-to]

Summary

Add explicit improve and verify stages to the automated task pipeline so task plans can be refined before implementation and completed work can be validated before it is considered done.

Current Flow

Tasks currently move through:

Backlog -> Planning -> Plan Ready -> Implementing -> Review -> Done

Proposed Flow

Extend the pipeline to include:

Backlog -> Planning -> Improve -> Plan Ready -> Implementing -> Review -> Verify -> Done

Expected Behavior

• Add a dedicated Improve stage after the initial plan is generated.
• The Improve stage should refine and strengthen the implementation plan before the task becomes Plan Ready.
• Add a dedicated Verify stage after review and before Done.
• The Verify stage should validate that the implementation satisfies the plan, compiles, and passes the relevant checks.
• Update the state machine so transitions are explicit and valid.
• Update the coordinator/subagent workflow routing so the new stages are processed automatically.
• Update task status types, API contracts, WebSocket events, and frontend Kanban columns as needed.
• Keep runtime adapter behavior neutral and route stage work through the existing runtime/profile resolution path.
• Update documentation to describe the new pipeline.
• Add or update tests for state transitions, coordinator behavior, API serialization, and UI stage rendering.

Acceptance Criteria

• Tasks can move from Planning to Improve, then to Plan Ready without manual database edits.
• The Improve stage updates/refines the plan artifact rather than implementation code.
• Tasks can move from Review to Verify, then to Done without manual database edits.
• Existing tasks remain compatible after migration or startup.
• The Kanban board shows the new stages in the correct order.
• Invalid stage transitions are rejected by the state machine.
• npm run ai:validate passes.