
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 precisely cut from the log following two planar cuts: one perpendicular to the axis of the cylinder, and another making a $45^\circ$ angle with the first. The intersection of these two planes has exactly one point in common with the log. Additionally, a conical cap with the same height as the wedge's height and base radius equal to the radius of the original cylindrical cut is placed on the second planar cut, making a combined solid. Find the volume of this new solid in terms of $\pi$, with $n$ as a positive integer.

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