
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 prism \(A B C A_1 B_1 C_1\) is circumscribed around a sphere with radius \(R\). Points \(M\) and \(N\) are the midpoints of the edges \(B B_1\) and \(C C_1\), respectively. A cylinder is inscribed in the sphere such that its base lies in the plane \(A M N\). Find the volume of the cylinder.

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