
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 right-angled triangle with area \( S \) and an acute angle \( \alpha \) is inscribed in the base of a spherical segment. Find the height of the segment if the corresponding central angle of its arc in axial section is equal to \( \beta \).

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