
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 cylinder is called equilateral if its height is equal to the diameter of its base. An equilateral cylinder is inscribed in a regular tetrahedron with edge length \(a\) such that one of the cylinder's bases lies on one of the faces of the tetrahedron, and the circle of the other base touches the other three faces. Calculate the volume and the lateral surface area of this cylinder.

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