
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
A cube is inscribed in a regular quadrilateral pyramid; the vertices of its upper base lie on the lateral edges, and the vertices of its lower base lie in the plane of the pyramid's base. Find the ratio of the volume of the cube to the volume of the pyramid if the lateral edge of the pyramid makes an angle $\alpha$ with the base plane.

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