
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 length of the base of a regular quadrangular pyramid is \(a\), and the dihedral angle at the base is \(\alpha\). A sphere is inscribed in this pyramid. Find the volume of the pyramid whose vertices are the points of tangency of the sphere with the lateral faces of the given pyramid and an arbitrary point lying in the plane of the base of the given pyramid.

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