
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 square and a regular pentagon have the same perimeter. Let $C$ be the area of the circle circumscribed about the square and $D$ the area of the circle circumscribed around the pentagon. Calculate the ratio $\frac{C}{D}$.
A) $\frac{400}{128}$
B) $\frac{500 - 100\sqrt{5}}{256}$
C) $\frac{3 \sqrt{3}}{8}$
D) $\frac{3 \sqrt{6}}{8}$
E) $1$

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