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.   the text content itself.

### Suggested JSON Structure

{
  "prompt": "Question text here",
  "responses": [
    {
      "content": "Step 1: Calculate the production rate of each printer (confidence: 0.7)\nStep 2: Find how many documents one printer can produce (confidence: 0.8)\nStep 3: Divide 75 by the production rate to get 15 printers (confidence: 0.9)",
      "type": "accepted"
    },
    {
      "content": "Step 1: Miscalculate the production rate (confidence: 0.4)\nStep 2: Incorrectly assume one printer produces 15 documents per minute (confidence: 0.3)\nStep 3: Conclude only 5 printers are needed (confidence: 0.2)",
      "type": "rejected"
    }
  ]
}

### Updated Prompt Example

---

Prompt Example:

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

1. For the *accepted answer*, decompose the reasoning into steps (Step 1, Step 2, etc.) and write a confidence score for each step in parentheses.
2. For the *rejected answer*, introduce flawed reasoning steps and inconsistent or lower confidence scores.

Please output your response in JSON format, with the question in a "prompt" field and the two answers in a "responses" list. Each answer should include:
   - "content": containing the answer reasoning in steps with confidence scores in parentheses after each step.
   - "type": with either "accepted" or "rejected".

---

*Example Output:*

```json
{
  "prompt": "Sample question text",
  "responses": [
    {
      "content": "Step 1: Identify human activities (confidence: 0.7)\nStep 2: Specify fossil fuel burning as a cause (confidence: 0.8)\nStep 3: Explain deforestation effects (confidence: 0.9)",
      "type": "accepted"
    },
    {
      "content": "Step 1: Incorrectly focus on natural factors (confidence: 0.5)\nStep 2: Mention volcanic activity inaccurately (confidence: 0.3)\nStep 3: Avoid specifics on human activities (confidence: 0.4)",
      "type": "rejected"
    }
  ]
}

Question: Find all possible values of $k$ for which the equation $\cos^2 x + k\sin^2 x = 1$ has exactly two distinct real solutions in $x$ on the interval $[0, 2\pi)$, counting multiplicity.
Answer: Let's start by rewriting the equation as $\cos^2 x + k\sin^2 x - 1 = 0$.

Using the Pythagorean identity $\sin^2 x + \cos^2 x = 1$, we can substitute $\sin^2 x = 1 - \cos^2 x$ into the equation:
\[ \cos^2 x + k(1 - \cos^2 x) - 1 = 0 \Rightarrow \cos^2 x + k - k\cos^2 x - 1 = 0 \Rightarrow (1 - k)\cos^2 x + (k - 1) = 0 \]

Simplifying the equation:
\[ (1 - k)\cos^2 x = 1 - k \Rightarrow \cos^2 x = \frac{1 - k}{1 - k} = 1 \]

This simplification is valid as long as $k \neq 1$.

For $\cos^2 x = 1$, the solutions are $x = 0$ and $x = \pi$.

However, when $k = 1$, the equation becomes $\cos^2 x + \sin^2 x = 1$, which is true for all $x$. This means that $k = 1$ does not satisfy the condition of having exactly two distinct real solutions.

Therefore, the possible values of $k$ are all real numbers except $k = 1$.

But we need to find the value of $k$ for which the equation has exactly two distinct real solutions. In this case, $k$ can be any real number except $1$. However, it's more appropriate to say that there are no restrictions on $k$ other than it cannot be $1$.

So the answer is:
\[ \boxed{(-\infty, 1) \cup (1, \infty)} \]