
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 20 inches and a height of 20 inches. A wedge is cut from the log with two planar cuts: the first is perpendicular to the axis of the cylinder, and the second forms a $30^\circ$ angle with the plane of the first cut. The intersection of these two planes has exactly one point in common with the log. Find the volume of the wedge expressed in the form $n \pi$, where $n$ is a positive number.

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