
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
Consider a line segment \( \overline{AB} \) with a length of \( 3r \). A point \( C \) lies on \( \overline{AB} \) such that \( AC = r \) and \( CB = 2r \). Construct semi-circles on diameters \( \overline{AB} \), \( \overline{AC} \), and \( \overline{CB} \) that are mutually tangent. The line \( \overline{CD} \) is perpendicular to \( \overline{AB} \) and intersects the semi-circle on \( \overline{CB} \) at \( D \). Find the ratio of the shaded area to the area of a circle with \( \overline{CD} \) as its radius.

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