
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 cone's base. A line passing through the center of the upper base of the cylinder and a point on the circumference of the cone's base makes an angle \(\alpha\) with the plane of the base. Find the ratio of the volumes of the cone and the cylinder if the angle between the slant height and the height of the cone is \(\beta\).

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