
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
Consider a square of side length  $1$  and erect equilateral triangles of side length  $1$  on all four sides of the square such that one triangle lies inside the square and the remaining three lie outside. Going clockwise around the square, let  $A$ ,  $B$ ,  $C$ ,  $D$  be the circumcenters of the four equilateral triangles. Compute the area of  $ABCD$ .

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