
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, each circle is divided into two equal areas, and $O$ is the center of the larger circle. The area of the larger circle is $100\pi$. Two smaller circles of equal size lie within the larger circle such that each smaller circle touches $O$ and the perimeter of the larger circle. What is the total area of the shaded regions? [asy]
size(100);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);fill(Arc((1,0),1,180,360)--cycle,mediumgray); fill(Arc((-1,0),1,180,360)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((1,0),1));
draw(Circle((-1,0),1));
dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1,-1)--(1,-1));
[/asy]

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