
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 given diagram, semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$ where point $C$ is no longer halfway between $A$ and $B$. Instead, let $\overline{AC} = 2x$ and $\overline{CB} = 6x$ such that $\overline{AB} = 8x$. All semi-circles are mutually tangent, and $\overline{CD} \perp \overline{AB}$. The radius of the first semicircle ($\overline{AB}$) is therefore $4x$, for $\overline{AC}$ it is $x$, and for $\overline{CB}$, it is $3x$. Calculate 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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