
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 surface area of a spherical cap is $n$ times larger than the lateral surface area of the cone that can be inscribed in the corresponding spherical slice. What is the ratio of the height of the slice to the diameter of the sphere?

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