
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 rests on a flat surface, pointing upward. The cross-section triangle, perpendicular to the base, has a vertex angle of 90 degrees. The diameter of the cone's base is $24$ inches. A sphere is placed inside the cone so that it is tangent to the sides of the cone and also rests on the flat surface. Determine the volume of this sphere in terms of $\pi$.

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