
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
In a regular triangular pyramid with the angle $\alpha$ between a lateral edge and a side of the base, a cross-section is made through the midpoint of a lateral edge parallel to the lateral face. Given the area $S$ of this cross-section, find the volume of the pyramid. What are the possible values of $\alpha$?

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