
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 dimensions $AB=4$ cm and $BC=2$ cm. A circle with center $B$ passes through $D$ forming arc $AFD$. A circle with center $D$ passes through $B$ forming arc $BEC$. Calculate the total area in square centimeters of regions II and III combined.

[asy]
path a=(13,7)..(20,0)--(0,20)..cycle;
path b=(7,13)..(20,0)--(0,20)..cycle;
draw(a);
draw(b);
dot((15.3,4.5));
label("F",(15.3,4.5),SE);
label("E",(4.7,15.5),NW);
draw((0,0)--(40,0)--(40,20)--(0,20)--cycle);
label("$A$",(0,0),SW);
label("$B$",(0,20),NW);
label("$C$",(40,20),NE);
label("$D$",(40,0),SE);
label("I",(2,19),S);
label("II",(22,13),S);
label("III",(18,7),N);
label("IV",(38,1),N);
[/asy]

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