
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 right triangle $ABC$ with $AB$ as the hypotenuse and line $BC$ as one of the other sides. A semicircle is constructed externally on each side of the triangle ($\overline{AB}$, $\overline{BC}$, and $\overline{CA}$) so that they are mutually tangent. The largest semicircle is on $\overline{AB}$, and the smallest on $\overline{CA}$. If $\overline{AB} = 8$, $\overline{BC} = 6$, and $\overline{CA} = 2\sqrt{7}$, find the ratio of the shaded area formed between the semicircles and the triangle to the area of a circle with $\overline{CA}$ as its diameter.

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