
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 has a radius of $7$ and a height of $10$. It is painted red all over. Points $P$ and $Q$ are positioned on the rim of the top circular face such that the arc $\overarc{PQ}$ measures $150^\text{o}$. The cylinder is then sliced along the plane passing through $P$, $Q$, and the cylinder's axis, revealing an unpainted cross-sectional face. Calculate the area of this face in the form of $d\cdot\pi + e\sqrt{f}$, where $d$, $e$, and $f$ are integers with $f$ not divisible by a square of any prime. Find $d+e+f$.

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