
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 a pyramid is a square with side length \(a\); two lateral faces of the pyramid are perpendicular to the base, and the larger lateral edge is inclined at an angle \(\beta\) to the base. A rectangular parallelepiped is inscribed in the pyramid; one of its bases lies in the plane of the pyramid's base, and the vertices of the other base are on the lateral edges of the pyramid. Find the volume of the parallelepiped if its diagonal makes an angle \(\alpha\) with the base plane.

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