
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
The angle between the height and the slant height of a cone is $\alpha$. A regular triangular prism is inscribed in the cone; the lower base of the prism lies in the plane of the base of the cone. The lateral faces of the prism are squares. Find the ratio of the lateral surface areas of the prism and the cone.

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