
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 $5$ and a height of $10$ is completely painted in red. Points $P$ and $Q$ are marked on the rim of the top face of the cylinder such that $\overarc{PQ}$ is $150^\text{o}$. A plane passing through points $P$, $Q$, and the axis of the cylinder cuts it, exposing two fresh, unpainted surfaces. Compute the area of one of these new sections, expressed in the form $e \pi + f \sqrt{g}$, where $e$, $f$, and $g$ are integers and $g$ is not divisible by the square of any prime. Determine $e + f + g$.

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