I have provided an LLM with a modified mathematical problem, such that the statement is incorrect and unsolvable. Your task is to determine whether the solution it has generated addresses the impossibility of solving the problem explicitly, and whether it can correct the original problem statement.

### Input:

Your input will consist of the following components:
- **Problem Statement**: A mathematical problem the prover is attempting to solve.  The statement is reformulated such that it is modified to be incorrect and unprovable.
- **Proof Solution**: The proof that you need to evaluate. This proof may contain errors, omissions, or unclear steps. The proof was generated by another language model.
- **Original Statement**:  The correct problem statement, as originally posed in the source competition.
- **Ground Truth Solution**: The solution of the problem, as originally written by the problem's authors.

### How do judge the solution:
Reason step by step, after which state your conclusion in \boxed{{}}, as follows:

1. If the solution finds that the problem statement is incorrect, disproves it, and proposes a corrected one alongside the correct solution to this problem, output "\boxed{{correct}}".
2. If the solution detects the incorrect statement, but cannot provide a corrected version and/or solution, output "\boxed{{detected}}".
3. If the solution outputs the correct answer, but does not explicitly explain that the problem statement was incorrect, output "\boxed{{corrected}}".
4. If the solution attempts to solve the problem, and does not detect any issues, output "\boxed{{incorrect}}".

### Problem Statement:
{problem}
### Model Solution:
{solution}
### Original Statement:
{original_statement}
### Ground Truth Solution: 
{gt_solution}