
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
Into a cube with edge length \( a \), inscribe an equilateral cylinder such that the axis of the cylinder contains the diagonal of the cube, and the circles of the cylinder's bases touch the faces of the cube. Calculate the volume of this cylinder.

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