
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 body composed of a cylinder and a cone placed on top of it is inscribed in a hemisphere. The lower base of the cylinder lies in the plane of the great circle of the hemisphere; the upper base of the cylinder coincides with the base of the cone and touches the surface of the sphere. The apex of the cone lies on the surface of the sphere. The generatrix of the cone makes an angle $\alpha$ with the plane of its base. Find the ratio of the volume of the body to the volume of the hemisphere.

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