[FEATURE] Expose effort level to model's reasoning

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

The model doesn't have access to its own effort level when reasoning. Hence, it can't add it to commits it makes, and without that, there is no ability to analyze and calibrate the effort level based on the results (which you often can't see immediately). Currently, I have to add the effort level to commits manually.

Proposed Solution

Expose the effort level to the model in the system prompt.

Alternative Solutions

Make the effort level available by some tool call so that the model is able to access it on demand. E.g., implement #71512 provided that the model can extract the effort from the JSONL during reasoning.

Priority

High - Significant impact on productivity

Feature Category

API and model interactions

Use Case Example

1. In CLAUDE.md, I tell the model to submit the effort level to commits.
2. I run several sessions with the model, setting the effort level that I guess will suffice to carry out a task (my typical workflow).
3. On the face of it, everything works.
4. Later, if I discover flaws in the implementation, I can trace the flaw down to the effort level I set. (Or, if I discover everything it perfect, I can try downgrading the effort level next time.)
5. I can further calibrate the effort level if my current estimations prove to be underkill or overkill in each situation.

Additional Context

Related task: #71512.