
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 cylinder with a radius of $8$ and a height of $10$ is painted red. Points $P$ and $Q$ are marked on the edge of one circular face such that the arc $\overarc{PQ}$ measures $150^\text{o}$. The cylinder is sliced along a plane passing through points $P$, $Q$, and the cylinder's center, revealing a new, unpainted face. Determine the area of this face in the form $d\pi + e\sqrt{f}$, where $d$, $e$, and $f$ are integers with $f$ 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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