You will receive a question-answer pair. Your task is to:

1. Decompose the answer into a sequence of numbered steps ("Step 1", "Step 2", etc.).
2. For each step, include a reasoning description and assign a confidence score (a float between 0 and 1).
3. For **accepted answers**, ensure reasoning steps are correct, with confidence scores that progressively increase as the steps approach the final answer.
4. For **rejected answers**, introduce flaws or errors in reasoning steps, and use inconsistent or lower confidence scores.

Please output the response in the JSON format below, following the example provided.


---

*Expected JSON Output Format:*

{
  "question": "[Insert question text here]",
  "accepted_answer": {
    "Step 1": {
      "reasoning": "[Describe the first reasoning step here]",
      "confidence_score": [Float between 0 and 1, e.g., 0.7]
    },
    "Step 2": {
      "reasoning": "[Describe the next reasoning step]",
      "confidence_score": [Incrementally higher float, e.g., 0.8]
    },
    "Step 3": {
      "reasoning": "[Continue describing each step with incrementally higher scores]",
      "confidence_score": [Approaching 1, e.g., 0.9]
    }
    // Continue until the final reasoning step
  },
  "rejected_answer": {
    "Step 1": {
      "reasoning": "[Introduce an incorrect or misleading reasoning step here]",
      "confidence_score": [Lower or inconsistent float, e.g., 0.5]
    },
    "Step 2": {
      "reasoning": "[Add a flawed or irrelevant reasoning step]",
      "confidence_score": [Inconsistent or lower float, e.g., 0.4]
    },
    "Step 3": {
      "reasoning": "[Continue with errors or inconsistencies in reasoning]",
      "confidence_score": [Non-incremental or lower float, e.g., 0.6]
    }
    // Continue until the final flawed reasoning step
  }
}

---

*Example JSON Output:*

{
  "question": "What are the primary causes of climate change?",
  "accepted_answer": {
    "Step 1": {
      "reasoning": "Identify human activities as significant contributors to climate change.",
      "confidence_score": 0.7
    },
    "Step 2": {
      "reasoning": "Specify fossil fuel burning as a major source of greenhouse gases.",
      "confidence_score": 0.8
    },
    "Step 3": {
      "reasoning": "Explain how deforestation reduces CO2 absorption, increasing atmospheric CO2.",
      "confidence_score": 0.9
    }
  },
  "rejected_answer": {
    "Step 1": {
      "reasoning": "Suggest natural factors as primary contributors to climate change.",
      "confidence_score": 0.5
    },
    "Step 2": {
      "reasoning": "Mention volcanic activity as a significant greenhouse gas source, incorrect for recent climate trends.",
      "confidence_score": 0.4
    },
    "Step 3": {
      "reasoning": "Briefly describe human activities without identifying specific sources.",
      "confidence_score": 0.6
    }
  }
}

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 $\boxed{\frac{19\sqrt{26}}{26}}$.