
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 cylindrical log has a diameter of $16$ inches. A wedge is cut from the log by making two planar cuts. The first cut is perpendicular to the axis of the cylinder, and the plane of the second cut forms a $30^\circ$ angle with the plane of the first cut. The intersection of these two planes still has exactly one point in common with the log. Determine the number of cubic inches in the wedge, expressed as $n\pi$, where $n$ is a positive integer.

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