
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 block of wood with a radius of $5$ and height $10$ is painted red. Points $P$ and $Q$ are selected on the boundary of one of the circular faces such that $\overarc{PQ}$ measures $90^\text{o}$. The cylinder is then sliced along the plane passing through points $P$, $Q$, and the cylinder's center, revealing a flat, unpainted face on each half. Find the area of one of these unpainted faces, expressed as $d\cdot\pi + e\sqrt{f}$, where $d$, $e$, and $f$ are integers with $f$ not divisible by the square of any prime. Calculate $d+e+f$.

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