
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 an isosceles triangle \( ABC \), where \( AB = BC \) and angle \( B \) is \( \frac{\pi}{4} \), a perpendicular \( AD \) is dropped onto side \( BC \). Semicircles are inscribed in triangles \( ABD \) and \( ADC \) such that their diameters lie on \( BD \) and \( AD \) respectively. Find the ratio of the areas of the constructed semicircles.

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