
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
In the diagram, three circles are arranged such that each has an area divided into two equal parts, and $O$ is the center of the largest circle. The area of the largest circle is $100\pi$. A second smaller circle is centered at a point on the diameter of the largest circle and touches the circumference of the largest circle. A third circle of radius equal to that of the second circle is centered at a point such that its circumference passes through the center $O$ of the largest circle and it intersects the second circle. Calculate the total area of the shaded regions in all three circles. [asy]
size(120);
import graph;
fill(Arc((0,0),3.16,180,360)--cycle,mediumgray);
fill(Arc((0,1.58),1.58,0,180)--cycle,mediumgray);
fill(Arc((1.58,0),1.58,90,270)--cycle,mediumgray);
draw(Circle((0,0),3.16));
draw(Circle((0,1.58),1.58));
draw(Circle((1.58,0),1.58));

dot((0,0)); label("$O$",(0,0),N);
draw((-3.16,0)--(3.16,0)); draw((-1.58,1.58)--(1.58,1.58));
[/asy]

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