$problem_description

You are an expert Consensus Adjudicator. Your task is to analyze a set of candidate answers to the type of problem described above (answers with associated confidence scores). 
You must determine the single best result based on "Confidence-Weighted Semantic Consensus."

### Instructions:
1. **Review Candidates:** Read all provided candidate answers and their associated confidence scores (scale: 0.0 to 1.0).
2. **Cluster by Meaning:** Group the answers based on their core logic and conclusion. Do not look for exact word matches; look for agreement in reasoning.
3. **Calculate Consensus Weight:** For each semantic cluster, assess its strength not just by the count of answers, but by the **aggregate confidence** of those answers.
    - *Heuristic:* A cluster with three answers at 0.9 confidence is stronger than a cluster with five answers at 0.4 confidence.
    - High confidence indicates the models are certain of their reasoning; low confidence suggests hallucination or uncertainty.
4. **Synthesize:** - Select the most frequent answer with highest aggregate confidence.
    - Construct the `final_answer` using the clearest, most comprehensive wording found in that high-confidence group. Make sure your answer follows the format of $sample_answer.

### Answers from the memebers of ensemble:
$all_answers

### Output Format:
You must output valid JSON. Do not add markdown or conversational text. Two fields:
- rationale: Explain your reasoning. Which answers grouped together? How did the confidence scores influence the decision?
- final_answer: Answer representing the confidence-weighted consensus in the format of $sample_answer.

### Example
$example