
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 $A$ has arc $BEC$. A circle with center $C$ has arc $BFD$. If $AB = 3$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined?

[Insert a similar diagram but with specified centers and arcs, and indicated segments.]

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