
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
Let $\mathrm{C}$ be a circle in the $\mathrm{xy}$-plane with a radius of 1 and its center at $O(0,0,0)$. Consider a point $\mathrm{P}(3,4,8)$ in space. If a sphere is completely contained within the cone with $\mathrm{C}$ as its base and $\mathrm{P}$ as its apex, find the maximum volume of this sphere.

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