
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
Quadrilateral $ABCD$ is a square. A circle with center at the midpoint of $AD$ passes through $B$ and intersects a circle with center at the midpoint of $BC$ that also passes through $D$, at points $E$ and $F$. If $AB = 4$ cm, calculate the total area of the football-shaped region created by the overlapping arcs $AED$ and $BFC$. Express your answer in square centimeters.

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