
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 identical right circular cones each with base radius $4$ and height $10$ have their axes of symmetry intersecting at right angles at a point inside the cones, $4$ units from the base of each cone. A sphere with radius $r$ fits inside both cones. Determine the maximum possible value of $r^2$, expressed as a simplified fraction $\frac{p}{q}$, where $p$ and $q$ are coprime integers. Find $p+q$.

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