
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 rectangle with $AB = 8$ and $BC = 4$. Let $E, F, G$ and $H$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the rectangle. What is the ratio of the area of square $EFGH$ to the area of rectangle $ABCD$?
A) $\frac{1}{2}$
B) $\frac{11 + 6\sqrt{3}}{16}$
C) $\sqrt{3}$
D) $\frac{11 + \sqrt{3}}{8}$

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