You will receive a question-answer pair and be asked to provide a response in JSON format. For each answer:

1. *For accepted answers*: Write the answer as a series of numbered steps (e.g., "Step 1", "Step 2", etc.). After each step, include a confidence score in parentheses, ranging from 0 to 1, where the score progressively increases toward the final answer.
   
2. *For rejected answers*: Include incorrect reasoning steps. Keep confidence scores inconsistent or lower to indicate flawed logic.

### JSON Output Structure

{
  "prompt": "[Question text]",
  "chosen": [
    {
      "content": "[Insert question text]",
      "role": "user"
    },
    {
      "content": "[Accepted answer as step-by-step reasoning, with confidence scores in parentheses after each step.]",
      "role": "assistant"
    }
  ],
  "rejected": [
    {
      "content": "[Insert question text]",
      "role": "user"
    },
    {
      "content": "[Rejected answer with flawed steps, each followed by a confidence score in parentheses.]",
      "role": "assistant"
    }
  ]
}

---

### Example Output

{
  "prompt": "How can I develop a habit of drawing daily?",
  "chosen": [
    {
      "content": "How can I develop a habit of drawing daily?",
      "role": "user"
    },
    {
      "content": "To build a daily drawing habit, try these steps:\n\nStep 1: Set aside a specific time each day for drawing to build consistency (confidence: 0.7).\nStep 2: Start with a short time commitment, like 5–10 minutes, to make it manageable (confidence: 0.8).\nStep 3: Track your progress to stay motivated as you see improvement over time (confidence: 0.9).\nStep 4: Gradually extend your drawing time as it becomes a natural part of your routine (confidence: 1.0).\n\nFollowing these steps can help establish a sustainable and rewarding daily drawing habit.",
      "role": "assistant"
    }
  ],
  "rejected": [
    {
      "content": "How can I develop a habit of drawing daily?",
      "role": "user"
    },
    {
      "content": "To build a drawing habit:\n\nStep 1: Draw only when feeling inspired, which may not be consistent (confidence: 0.4).\nStep 2: Set high standards for each drawing session, aiming for completed artwork (confidence: 0.3).\nStep 3: Take breaks for a few days if you feel tired, to avoid burnout (confidence: 0.2).\n\nThis approach may not support a daily habit, as it relies on sporadic inspiration rather than consistency.",
      "role": "assistant"
    }
  ]
}


Question: In isosceles triangle $\triangle ABC$ we have $AB=AC=4$.  The altitude from $B$ meets $\overline{AC}$ at $H$.  If $AH=3(HC)$ then determine $BC$.

Another problem similar to this one:

In isosceles triangle $\triangle ABC$ we have $AB=AC=5$.  The altitude from $B$ meets $\overline{AC}$ at $H$.  If $AH=2(HC)$ then determine $BC$.
Answer: Let's denote the length of side $BC$ as $a$, the length of side $AC$ as $b$, and the length of side $AB$ as $c$.
In our case $c = b = 4$, and we need to find $a$.

Since $\triangle ABC$ is isosceles, we know that $BH$ is perpendicular to $AC$.
Using the Pythagorean theorem in $\triangle ABH$ and $\triangle BCH$:
\[ c^2 = (BH)^2 + (AH)^2 \quad \text{and} \quad a^2 = (BH)^2 + (HC)^2 \]

It is given that $AH = 3(HC)$. Let $HC = x$, then $AH = 3x$. Since $AH + HC = AC$, we have $3x + x = b$.
\[ 3x + x = b \Rightarrow 4x = b \Rightarrow x = b : 4 \Rightarrow x = 1 \]

Now we can find $AH$:
\[ AH = 3x = 3(1) = 3 \]

Since $c = b = 4$ and $AH = 3$, we can find $BH$:
\[ c^2 = (BH)^2 + (AH)^2 \Rightarrow (BH)^2 = c^2 - (AH)^2 = 4^2 - 3^2 = 16 - 9 = 7 \Rightarrow BH = \sqrt{7} \]

Finally, we can find $a$:
\[ a^2 = (BH)^2 + (HC)^2 = (\sqrt{7})^2 + 1^2 = 7 + 1 = 8 \Rightarrow a = \sqrt{8} = \boxed{2\sqrt{2}} \]