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: The graph of the equation $y=ax^2+bx+c$ intersects the $x$-axis at two points, $P$ and $Q$. If the vertex of the parabola is equidistant from $P$ and $Q$, what is the relationship between the coefficients $a$, $b$, and $c$?
Answer: To find the relationship between the coefficients $a$, $b$, and $c$, we need to use the properties of the parabola.

The $x$-intercepts of the parabola are the solutions to the equation $ax^2 + bx + c = 0$. Let's denote these solutions as $x_1$ and $x_2$. The sum of the roots is $x_1 + x_2 = -\frac{b}{a}$, and the product of the roots is $x_1x_2 = \frac{c}{a}$.

The vertex of the parabola is the point $(h, k)$, where $h = -\frac{b}{2a}$ and $k = c - \frac{b^2}{4a}$.

Since the vertex is equidistant from the $x$-intercepts, we have:
\[ |h - x_1| = |h - x_2| \Rightarrow h - x_1 = x_2 - h \text{ or } h - x_1 = h - x_2 \]

Substitute $h = -\frac{b}{2a}$:
\[ -\frac{b}{2a} - x_1 = x_2 + \frac{b}{2a} \text{ or } -\frac{b}{2a} - x_1 = -\frac{b}{2a} - x_2 \]

Simplify the equations:
\[ x_1 + x_2 = -\frac{b}{a} \text{ or } x_1 = x_2 \]

Since $x_1$ and $x_2$ are the roots of the quadratic equation, we know that $x_1 + x_2 = -\frac{b}{a}$.

However, the condition $x_1 = x_2$ implies that the parabola has a double root, which means that the discriminant $b^2 - 4ac = 0$.

Therefore, the relationship between the coefficients $a$, $b$, and $c$ is:
\[ b^2 - 4ac = 0 \Rightarrow b^2 = 4ac \]

So, the answer is:
\[ \boxed{b^2 = 4ac} \]