
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 base of a triangular pyramid is an isosceles triangle with an area of \( S \) and a vertex angle of \( \alpha \). Find the volume of the pyramid if the angle between each lateral edge and the height of the pyramid is \( \beta \).

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