
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
Let $ABCD$ be a square with each side of 4 units. Construct equilateral triangles $ABE$, $BCF$, $CDG$, and $DAH$ inscribed on each side of the square, inside the square. Let $E, F, G, H$ be the centers, respectively, of these equilateral triangles. What is the ratio of the area of square $EFGH$ to the area of square $ABCD$?
A) $\frac{2}{3}$
B) $\frac{1}{2}$
C) $\frac{1}{3}$
D) $\sqrt{3}$
E) $\sqrt{2}$

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