
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 cylinder is inscribed in a cone; the lower base of the cylinder lies in the plane of the base of the cone. A line passing through the center of the upper base of the cylinder and a point on the circumference of the base of the cone makes an angle $\alpha$ with the base plane. Find the ratio of the volumes of the cone and the cylinder if the angle between the slant height and the altitude of the cone is $\beta$.

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