
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, and let $E, F, G, H$ be the midpoints of the hypotenuse of right-angled triangles with one leg as $\overline{AB}, \overline{BC}, \overline{CD}, \overline{DA}$ respectively, each exterior to the square. Given that each triangle has legs equal to the side of the square, find the ratio of the area of square $EFGH$ to the area of square $ABCD$.

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