
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 cylinder has a radius of $8$ cm and a height of $10$ cm. The entire surface of the cylinder is painted red. Points $P$ and $Q$ are positioned on the edge of one of the circular bases such that the arc $\overarc{PQ}$ on that face measures $90^\text{o}$. The cylinder is then cut along the plane that includes points $P$, $Q$, and the cylinder's axis, producing two new flat surfaces. Determine the area of one of these new flat surfaces in the form $x\pi + y\sqrt{z}$, where $x$, $y$, and $z$ are integers, and $z$ is not divisible by the square of any prime. Find $x+y+z$.

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