
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 the right prism \(A B C A_{1} B_{1} C_{1}\) \((A A_{1} \parallel B B_{1} \parallel C C_{1})\), the sides of the base \(A B\) and \(B C\) are \(a\) and \(b\) respectively, and the angle between them is \(\alpha\). A plane is drawn through the bisector of this angle and vertex \(A_{1}\), making an acute angle \(\beta\) with the plane of the base. Find the area of the cross-section.

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