
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 square with side length \(a\) is inscribed in the base of a cone. A plane passing through one of the sides of this square and the apex of the cone intersects the surface of the cone, forming an isosceles triangle with the apex angle \(\alpha\). Find the volume of the cone.

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