
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
In a square arrangement, a central square of side length $\sqrt{2}$ is surrounded by four regular pentagons, each with a side length equal to the square's side. What is the area of $\triangle ABC$, formed by connecting the centers of three consecutive surrounding pentagons?
A) $\frac{\sqrt{3}}{2 \sin^2(\pi/6)}$
B) $\frac{\sqrt{3}}{2 \sin^2(\pi/5)}$
C) $\frac{\sqrt{3}}{\sin^2(\pi/4)}$
D) $\frac{3\sqrt{3}}{4 \sin^2(\pi/5)}$

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