
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
The base of the pyramid \( FABC \) is an isosceles triangle \( ABC \), where the angle between the equal sides \( AB \) and \( AC \) is \( \alpha \) (\( \alpha < \pi / 2 \)). A triangular prism \( AED A_{1} E_{1} D_{1} \) is inscribed in the pyramid such that the points \( A_{1}, E_{1} \), and \( D_{1} \) lie on the lateral edges \( AF, CF \), and \( BF \) of the pyramid, respectively, and the side \( ED \) of the base \( AED \) passes through the center of the circumcircle of the triangle \( ABC \). Find the ratio of the volume of the prism to the volume of the pyramid.

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