
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 sits upright on a table, with a cross-section perpendicular to the base forming an isosceles right triangle. The triangle's vertex angle is \(90^\circ\), and the diameter of the cone's base is 12 inches. A sphere is placed inside the cone such that it is tangent to the sides of the cone and rests on the table. Compute 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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