
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 sphere is inscribed in a right circular cone, and the surface area of the sphere is $\frac{2}{3}$ of the lateral surface area of the cone. What is the aperture angle of the cone? (The aperture angle of the cone is the angle between two opposite generatrices of the right circular cone.)

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