
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
From the base of the height of a regular triangular pyramid, a perpendicular equal to \( r \) is dropped onto a lateral edge. Find the volume of the pyramid if the dihedral angle between the lateral face and the base of the pyramid is \( \alpha \).

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