
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 $D$ has arc $AEC$. A circle with center $B$ has arc $AFC$. A third circle with center $C$ has arc $AGC$. If $AB = 4$ cm, calculate the combined area of the overlapping regions formed by these three circles within the square. Express your answer as a decimal to the nearest tenth.

[asy]
path a=(7,13)..(0,0)--(40,40)..cycle;
path b=(13,7)..(0,0)--(40,40)..cycle;
path c=(27,33)..(0,0)--(40,40)..cycle;
draw(a);
draw(b);
draw(c);
dot((16,2));
label("F",(16,2),SE);
dot((24,38));
label("E",(24,38),NW);
dot((35,20));
label("G",(35,20),E);
draw((0,0)--(40,0)--(40,40)--(0,40)--cycle);
label("$A$",(0,0),SW);
label("$B$",(0,40),NW);
label("$C$",(40,40),NE);
label("$D$",(40,0),SE);
label("I",(4,38),S);
label("II",(18,26),S);
label("III",(22,14),N);
label("IV",(36,4),N);
[/asy]

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