
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 block of wood has the shape of a right circular cylinder with a radius of $8$ and a height of $10$. The entire surface of the block is painted red. Points $P$ and $Q$ are chosen on the edge of one of the circular faces such that the arc $\overarc{PQ}$ measures $180^\text{o}$. The block is then sliced in half along the plane passing through points $P$, $Q$, and the center of the cylinder, revealing a flat, unpainted face on each half. Determine the area of one of these unpainted faces expressed in terms of $\pi$ and $\sqrt{d}$. Find $a + b + d$ where the area is expressed as $a \pi + b\sqrt{d}$, with $a$, $b$, $d$ as integers, and $d$ not being divisible by the square of any prime.

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