
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
Through the altitude of an equilateral triangle with side length \(a\), a plane is drawn perpendicular to the plane of the triangle, and within this plane a line \(l\) is taken parallel to the altitude of the triangle. Find the volume of the solid obtained by rotating the triangle around the line \(l\).

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