
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 solid cylinder has a radius of 5 and a height of 10. It's completely painted red. Two points, P and Q, are placed on the top circular face of the cylinder such that the arc length $\overarc{PQ}$ measures $90^\text{o}$. The cylinder is then cut along a plane passing through P, Q, and the axis of the cylinder to reveal two new flat, unpainted faces. Determine the area of one of these new unpainted faces in the form $d\pi + e\sqrt{f}$, where $d$, $e$, and $f$ are integers and $f$ is not divisible by the square of any prime. Find $d+e+f$.

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