
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 circular cone is placed with its vertex pointing upward. The cone's axis is vertical, and a vertical cross-section through the axis forms an isosceles right triangle. The diameter of the cone's base is 12 inches. Inside the cone, a sphere is placed such that it touches the sides of the cone and sits on the base. Find the volume of the sphere in cubic inches in terms of $\pi$.

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