
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 pyramid is a right triangle with a hypotenuse equal to \(c\) and the smaller acute angle equal to \(\alpha\). The largest lateral edge makes an angle \(\beta\) with the plane of the base. Find the volume of the pyramid if its height passes through the intersection point of the medians of the base.

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