
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  $3$  and height  $8$  have axes of symmetry that intersect at right angles at a point in the interior of the cones a distance  $3$  from the base of each cone. A sphere with radius  $r$  lies inside both cones. The maximum possible value for  $r^2$  is  $\frac mn$ , 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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