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: A right rectangular prism has a height of 3 units and a square base with side length 4 units. A point P is chosen on the top face of the prism, such that the distance from P to the nearest edge of the prism is 1 unit. What is the length of the longest diagonal of the prism that passes through point P?
Answer: Let's visualize the prism and point $P$. Since the prism has a square base with side length 4, the diagonal of the base is $\sqrt{4^2 + 4^2} = 4\sqrt{2}$ units.

Point $P$ is 1 unit away from the nearest edge. This means that the point $P$ is on the top face, 1 unit away from one of the edges of the square base.

To find the longest diagonal passing through $P$, we need to consider the space diagonal of the prism. The space diagonal is the longest diagonal that can be drawn in the prism.

The space diagonal can be found using the 3D distance formula or by applying the Pythagorean theorem twice.

First, find the diagonal of the base that includes the point $P$. Since $P$ is 1 unit away from one edge of the square base, the diagonal of the base that includes $P$ will be the hypotenuse of a right triangle with legs 4 (the side of the square) and 3 (4 - 1 = 3, because $P$ is 1 unit away from the edge). This diagonal is $\sqrt{4^2 + 3^2} = 5$ units.

Now, we have a right triangle with legs 5 (the diagonal of the base including $P$) and 3 (the height of the prism). The space diagonal (hypotenuse) is $\sqrt{5^2 + 3^2} = \sqrt{25 + 9} = \sqrt{34}$ units.

The answer is:
\[ \boxed{\sqrt{34}} \]