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: Problem:
In a cryptographic system, a message is encrypted using the formula:

C ≡ M^e (mod n)

where C is the ciphertext, M is the plaintext, e is the public key, and n is the modulus. Given that e = 17, n = 323, and C = 179, find the original plaintext message M, which is an integer from 0 to 322.

Note: You may need to use the properties of modular arithmetic and the concept of multiplicative inverses to solve this problem.
Answer: To find the original plaintext message M, we need to find the value of M that satisfies the equation:

179 ≡ M^17 (mod 323)

Since 17 and 323 are relatively prime (gcd(17, 323) = 1), we can use the property of modular arithmetic that states:

a ≡ b (mod n) ⇒ a^k ≡ b^k (mod n)

for any integer k.

In this case, we can rewrite the equation as:

179 ≡ M^17 (mod 323)
⇒ 179^k ≡ (M^17)^k (mod 323)
⇒ 179^k ≡ M^(17k) (mod 323)

We want to find a value of k such that 17k ≡ 1 (mod 323), which is equivalent to finding the multiplicative inverse of 17 modulo 323.

Using the Extended Euclidean Algorithm, we can find that the multiplicative inverse of 17 modulo 323 is 19, since:

17 × 19 ≡ 1 (mod 323)

Therefore, we can rewrite the equation as:

179^19 ≡ M^(17 × 19) (mod 323)
⇒ 179^19 ≡ M^1 (mod 323)
⇒ 179^19 ≡ M (mod 323)

Now, we can compute 179^19 modulo 323 to find the value of M:

179^19 ≡ 97 (mod 323)

Thus, the original plaintext message M is:

M = 97

So, the answer is:

M = \boxed{97}