
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 have a base radius of $5$ and a height of $12$. The axes of symmetry for these cones intersect at right angles at a point in the interior of the cones, a distance of $4$ from the base of each cone. A sphere with radius \( r \) is positioned such that it fits within both cones. Determine the maximum possible value of \( r^2 \), and express it as a fraction \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m+n \).

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