
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]


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The side of an equilateral triangle is equal to \(a\). The triangle rotates around a line that lies in the plane of the triangle outside it, passing through the vertex of the triangle and forming an angle \(\alpha\) with the side. Find the volume of the solid of revolution and determine the value of \(\alpha\) at which this volume is maximized.

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