
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 right prism \(A B C A_{1} B_{1} C_{1}\) (where \(A A_{1} \parallel B B_{1} \parallel \mathrm{CC}_{1}\)) is an isosceles triangle \(A B C\) with \(A B = A C\). The perimeter of the triangle is \(2 p\), and the angle at vertex \(A\) is \(\alpha\). A plane passes through side \(B C\) and vertex \(A_{1}\), forming an angle \(\beta\) with the base plane. Find the volume of the prism.

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