
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 an isosceles acute-angled triangle with a base length $a$ and the opposite angle \(\alpha\). The lateral edge of the pyramid passing through the vertex of this angle forms an angle \(\beta\) with the plane of the base. Find the volume of the pyramid if the height of the pyramid passes through the intersection point of the heights of the base.

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