
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 table, 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 $18$ inches. A sphere is placed inside the cone such that it touches the sides of the cone and rests on the table. Determine the volume, in cubic inches, of the sphere, expressed in terms of $\pi$.

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