
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, semicircles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, such that they are mutually tangent. Point $C$ is at one third the distance from $A$ to $B$, so $\overline{AC} = \frac{1}{3} \overline{AB}$ and $\overline{CB} = \frac{2}{3} \overline{AB}$. If $\overline{CD} \perp \overline{AB}$, find the ratio of the shaded area to the area of a circle with $\overline{CD}$ as radius.

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