
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Two congruent right circular cones each with base radius $4$ and height $10$ have their axes of symmetry intersecting at right angles at a point in the interior of the cones a distance $4$ from the base of each cone. A sphere with radius $r$ is situated within both cones. Find the maximum possible value of $r^2$ expressed as a fraction $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Determine the sum $m+n$.

Present the answer in LaTex format: \boxed{Your answer}
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