
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
[asy] fill(circle((4,0),4),grey); fill((0,0)--(8,0)--(8,-4)--(0,-4)--cycle,white); fill(circle((6,0),2),white); fill(circle((2,0),2),white); draw((0,0)--(8,0),black+linewidth(1)); draw((4,0)--(4,sqrt(16)),black+linewidth(1)); MP("A", (0,0), W);  MP("B", (8,0), E);  MP("C", (4,0), S); MP("D",(4,sqrt(16)), N); [/asy]
In this diagram, semicircles are constructed on diameters \( \overline{AB} \), \( \overline{AC} \), and \( \overline{CB} \), such that they are mutually tangent. If \( \overline{CD} \bot \overline{AB} \), and \( \overline{AC} = \overline{CB} \), then 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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