
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 \(5\) and height \(10\) have the axes of symmetry that intersect at right angles at a point in the interior of the cones a distance \(5\) from the base of each cone. A sphere with radius \(r\) lies within both cones. Calculate the square of the maximum possible value of \(r\) and express the result as \(\frac{m}{n}\), where \(m\) and \(n\) are relatively prime integers. Find \(m+n\).

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