
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 side of the square at the base of a regular quadrilateral pyramid is equal to \( a \). A regular quadrilateral prism is inscribed in the pyramid; the vertices of the upper base lie on the lateral edges, and the vertices of the lower base lie in the plane of the pyramid's base. The diagonal of the prism forms an angle \( \varphi \) with the plane of the base. Find the volume of the prism if the lateral edge of the pyramid forms an angle \( \alpha \) with the plane of the base.

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