
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 sculpture with a radius of $7$ and height of $9$ is painted green. Two points, $E$ and $F$, are positioned on the rim of one of its circular bases, such that the arc $\overarc{EF}$ measures $150^\text{o}$. The cylinder is sliced through plane passing through points $E$, $F$, and the cylinder's center, unveiling two fresh, unpainted surfaces. Determine the area of one of these new surfaces in the form $p\pi + q\sqrt{r}$, where $p$, $q$, $r$ are integers, and $r$ is not divisible by the square of any prime. Find the sum $p+q+r$.

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