
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
A right triangle  $ABC$  is inscribed in the circular base of a cone. If two of the side lengths of  $ABC$  are  $3$  and  $4$ , and the distance from the vertex of the cone to any point on the circumference of the base is  $3$ , then the minimum possible volume of the cone can be written as  $\frac{m\pi\sqrt{n}}{p}$ , where  $m$ ,  $n$ , and  $p$  are positive integers,  $m$  and  $p$  are relatively prime, and  $n$  is squarefree. Find  $m + n + p$ .

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