
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 the largest circle, with center \(O\), has an area of \(100\pi\). Each circle is divided into two equal areas. The smallest circle is entirely inside the largest, touching it at one point, and the medium circle touches both the smallest and largest circles at their respective borders. Calculate the total area of the shaded regions. 
[asy]
size(100);
import graph;
fill(Arc((0,0),3.16,180,360)--cycle,mediumgray);fill(Arc((0,1.5),1.58,0,180)--cycle,mediumgray); fill(Arc((0,-0.79),0.79,180,360)--cycle,mediumgray);
draw(Circle((0,0),3.16));
draw(Circle((0,1.5),1.58));
draw(Circle((0,-0.79),0.79));

dot((0,0)); label("$O$",(0,0),N);
draw((-3.16,0)--(3.16,0)); draw((-1.58,1.5)--(1.58,1.5)); draw((-0.79,-0.79)--(0.79,-0.79));
[/asy]

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