
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
Rectangle $ABCD$ has sides $AB = 3$ cm and $BC = 4$ cm. A circle with center $A$ has arc $BEC$, and a circle with center $C$ has arc $BFD$. Calculate the total number of square centimeters in the football-shaped area of regions II and III combined, assuming similar labeling and setups to the original problem, highlighting overlapping regions of two quarter-circles associated with diagonals of the rectangle.

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