
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 triangular pyramid is \(a\), and the dihedral angle at the base is \(\alpha\). An inscribed right triangular prism is given such that three of its vertices lie on the slant heights (or lateral edges) of the pyramid, and the other three vertices lie in the plane of the base of the pyramid. Find the volume of the prism, given that the center of the sphere inscribed in the pyramid lies in the plane of the upper base of the prism.

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