
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 height of $10$ units and a radius of $7$ units. This cylinder is painted red on its surface. Two points, $P$ and $Q$, are chosen on the perimeter of one base such that the minor arc $\overarc{PQ}$ subtends an angle of $150^\text{o}$ at the center. The cylinder is then cut through a plane passing through $P$, $Q$, and the axis of the cylinder, revealing two unpainted cross-sections. Calculate the area of one of these cross-sections, expressed in the form $d\cdot\pi + r\sqrt{s}$, where $d$, $r$, and $s$ are integers with $s$ not divisible by the square of any prime. Determine $d+r+s$.

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