
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 vertices $A(0,0)$, $B(a,0)$, $C(a,a)$, and $D(0,a)$. At each side of the square, construct an isosceles right-angled triangle with the hypotenuse as the side of the square and exterior to the square. Let $E, F, G,$ and $H$ be the centroids of these triangles along sides $AB, BC, CD,$ and $DA$ respectively. Find the ratio of the area of square $EFGH$ to the area of square $ABCD$.
A) $\frac{2 + \sqrt{2}}{9}$
B) $\frac{2}{3}$
C) $\frac{2\sqrt{2}}{3}$
D) $\frac{3 + \sqrt{2}}{6}$
E) $2 + \sqrt{2}$

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