
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 $PQRS$ is a square. A circle with center $Q$ has arc $PTU$. A circle with center $S$ has arc $PVW$. If $PQ = 4$ cm, what is the total number of square centimeters in the football-shaped area of regions II and III combined?

[asy]
path a=(7,13)..(0,0)--(40,40)..cycle;
path b=(33,7)..(0,0)--(40,40)..cycle;
draw(a);
draw(b);
dot((17,7));
label("W",(17,7),SE);
label("T",(23,33),NW);
dot((23,33));
draw((0,0)--(40,0)--(40,40)--(0,40)--cycle);
label("$P$",(0,0),SW);
label("$Q$",(0,40),NW);
label("$R$",(40,40),NE);
label("$S$",(40,0),SE);
label("I",(4,38),S);
label("II",(19,26),S);
label("III",(21,14),N);
label("IV",(36,2),N);
[/asy]

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