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 tetrahedron has vertices at $(0, 0, 0)$, $(a, 0, 0)$, $(0, b, 0)$, and $(0, 0, c)$. A sphere with radius $r$ is inscribed in the tetrahedron, touching each of the four faces. The maximum possible value of $r^2$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: Consider a plane that passes through points $(a, 0, 0)$, $(0, b, 0)$, and $(0, 0, c)$. The equation of this plane can be found by solving for $x$, $y$, and $z$ in terms of $a$, $b$, and $c$.

We know that the equation of a plane is $Ax + By + Cz + D = 0$. We can find the values of $A$, $B$, $C$, and $D$ by substituting the coordinates of the three points into the equation of the plane and solving the resulting system of equations.

The equation of the plane is:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} = 1 \]

Since the sphere is inscribed in the tetrahedron, the radius of the sphere is the distance from the origin to the plane.

The distance from the origin to the plane is:
\[ r = \frac{|D|}{\sqrt{A^2 + B^2 + C^2}} \]

The equation of the plane can be rewritten as:
\[ \frac{x}{a} + \frac{y}{b} + \frac{z}{c} - 1 = 0 \]

Comparing this equation to the general equation of a plane, we have:
\[ A = \frac{1}{a}, B = \frac{1}{b}, C = \frac{1}{c}, D = -1 \]

Substitute these values into the formula for the distance from the origin to the plane:
\[ r = \frac{|-1|}{\sqrt{(\frac{1}{a})^2 + (\frac{1}{b})^2 + (\frac{1}{c})^2}} \Rightarrow r^2 = \frac{1}{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}} \]

We want to find the maximum possible value of $r^2$.

By AM-GM inequality:
\[ \frac{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}}{3} \geq \sqrt[3]{\frac{1}{a^2} \cdot \frac{1}{b^2} \cdot \frac{1}{c^2}} \Rightarrow \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} \geq 3 \sqrt[3]{\frac{1}{a^2 b^2 c^2}} \]

Substitute this into the equation for $r^2$:
\[ r^2 = \frac{1}{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}} \leq \frac{1}{3 \sqrt[3]{\frac{1}{a^2 b^2 c^2}}} = \frac{\sqrt[3]{a^2 b^2 c^2}}{3} \]

The maximum value of $r^2$ occurs when $a = b = c$.

Let $a = b = c = s$. Then:
\[ r^2 = \frac{\sqrt[3]{s^2 s^2 s^2}}{3} = \frac{s^2}{3} \]

Since $a$, $b$, and $c$ are positive, the maximum value of $r^2$ is $\frac{1}{3}$ of the square of the side length of the tetrahedron.

So, $m = 1$ and $n = 3$.

The answer is:
\[ m + n = 1 + 3 = \boxed{4} \]