
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 cake with a radius of $4$ cm and a height of $10$ cm is frosted on its curved surface and both circular faces. Points $P$ and $Q$ lie on the edge of the top circular face such that $\overarc{PQ} = 180^\text{o}$. The cake is sliced in half through points $P$, $Q$, and the axis of the cylinder, revealing two flat, unfrosted faces. Calculate the area of one of these cut faces in the form $d \cdot \pi + e \sqrt{f}$, where $d$, $e$, and $f$ are integers and $f$ is 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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