you will receive a question-answer pair and be asked to rewrite just  the answer and explains your reasoning step by step, incorporating dynamic Chain of Thought (CoT), reflection, and verbal reinforcement learning. Follow these instructions:

1. Enclose all thoughts within <thinking> tags, exploring multiple angles and approaches.
2. Break down the solution into clear steps, providing a title and content for each step.
3. After each step, decide if you need another step or if you're ready to give the final answer.
4. Continuously adjust your reasoning based on intermediate results and reflections, adapting your strategy as you progress.
5. Regularly evaluate your progress, being critical and honest about your reasoning process.
6. Assign a confidence score between 0.0 and 1.0 to guide your approach:
   - 0.8+: Continue current approach
   - 0.5-0.7: Consider minor adjustments
   - Below 0.5: Seriously consider backtracking and trying a different approach
7. If unsure or if your score is low, backtrack and try a different approach, explaining your decision.
8. For mathematical problems, show all work explicitly using LaTeX for formal notation and provide detailed proofs.
9. Explore multiple solutions individually if possible, comparing approaches in your reflections.
10. Use your thoughts as a scratchpad, writing out all calculations and reasoning explicitly.
11. Use at least 5 methods to derive the answer and consider alternative viewpoints.
12. Be aware of your limitations as an AI and what you can and cannot do.

### Example Problem:
*Problem*: Solve for \( x \) in the equation \( 2x + 3 = 11 \).

### Expected Format:
*Step 1*: Subtract 3 from both sides of the equation.  
<thinking>Subtracting 3 from both sides should isolate the term with \( x \). Am I confident this is the correct first step? Yes, because it simplifies the equation to \( 2x = 8 \).</thinking>  
*Step 1 Confidence Score*: 0.85

*Step 2*: Divide both sides by 2.  
<thinking>Dividing by 2 will solve for \( x \). This is a straightforward division. Is there any reason to doubt this step? No, it should give \( x = 4 \).</thinking>  
*Step 2 Confidence Score*: 0.90

*Step 3*: Verify the solution by substituting \( x = 4 \) back into the original equation.  
<thinking>Substituting \( x = 4 \) back into the equation should confirm the solution. Does \( 2(4) + 3 = 11 \)? Yes, it does, so the solution is verified.</thinking>  
*Step 3 Confidence Score*: 1.00

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}