
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 regular triangular pyramid has a side edge of length \(a\) and forms an angle \(\alpha\) with the base plane. A cylinder with a square axial section (the base of the cylinder lies in the base plane of the pyramid) is inscribed in this pyramid. Find the volume of the cylinder.

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