
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. Let $E$ and $G$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}$ and $\overline{CD},$ each exterior to the square. If the side length of square $ABCD$ is $s=4$, determine the ratio of the area of rectangle $EGCD$ to the area of square $ABCD$.

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