
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 largest circle. The area of the largest circle is $100\pi$. There are two smaller concentric circles with center $O$, their diameters being $2$ units and $4$ units less than the diameter of the largest circle. What is the total area of the shaded regions, where the shaded regions are the lower halves of the largest and smallest circles and the upper half of the middle circle?
[asy]
size(100);
import graph;
fill(Arc((0,0),2,180,360)--cycle,mediumgray);
fill(Arc((0,0),1.3,0,180)--cycle,mediumgray);
fill(Arc((0,0),1.8,180,360)--cycle,mediumgray);
draw(Circle((0,0),2));
draw(Circle((0,0),1.8));
draw(Circle((0,0),1.3));
dot((0,0)); label("$O$",(0,0),N);
draw((-2,0)--(2,0)); draw((-1.8,0)--(1.8,0)); draw((-1.3,0)--(1.3,0));
[/asy]

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