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: 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}