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: What is the length of the altitude dropped from the vertex $A$ of the triangle $ABC$ with vertices $A(0,3)$, $B(4,0)$, and $C(-1,-1)$ to the line $BC$?
Answer: The length of the altitude from vertex $A$ to line $BC$ can be found using the formula:
\[ d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
where $(x, y)$ is a point on the line and $Ax + By + C = 0$ is the equation of the line in standard form.

To find the equation of the line $BC$, we can use the slope-intercept form $y = mx + b$.

First, find the slope $m$ using the coordinates of $B$ and $C$:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-1)}{4 - (-1)} = \frac{1}{5} \]

Next, find the y-intercept $b$ using the coordinates of $B$ and the slope $m$:
\[ 0 = \frac{1}{5}(4) + b \Rightarrow b = -\frac{4}{5} \]

Now we have the equation of the line in slope-intercept form:
\[ y = \frac{1}{5}x - \frac{4}{5} \]

Convert this to the standard form $Ax + By + C = 0$:
\[ 5y = x - 4 \Rightarrow x - 5y - 4 = 0 \]

Now we can use the formula to find the length of the altitude:
\[ d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} = \frac{|(0) - 5(3) - 4|}{\sqrt{1^2 + (-5)^2}} = \frac{|-19|}{\sqrt{26}} = \frac{19}{\sqrt{26}} \]

Rationalize the denominator:
\[ \frac{19}{\sqrt{26}} = \frac{19}{\sqrt{26}} \times \frac{\sqrt{26}}{\sqrt{26}} = \frac{19\sqrt{26}}{26} \]

Therefore, the length of the altitude is $\boxed{\frac{19\sqrt{26}}{26}}$.