
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 first circle. The area of the first (largest) circle is $100\pi$. A second smaller circle touches the inside of the largest circle at one point, with its center positioned on the diameter of the largest. Introduce a third circle, smaller than the second, with its center on the circumference of the second circle and just touching the inside of the largest circle. Determine the total area of the shaded regions for all circles. [asy] size(120); import graph; fill(Arc((0,0),3,180,360)--cycle,mediumgray); fill(Arc((0,1.5),1.5,0,180)--cycle,mediumgray); fill(Arc((1.5,1.5),0.75,180,360)--cycle,mediumgray); draw(Circle((0,0),3)); draw(Circle((0,1.5),1.5)); draw(Circle((1.5,1.5),0.75)); dot((0,0)); label("$O$",(0,0),N); draw((-3,0)--(3,0)); draw((-1.5,1.5)--(1.5,1.5)); draw((1.5,0.75)--(2.25,1.5)); [/asy]

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