
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
Let \( AB \) be the diameter of the lower base of a cylinder, and \( A_1B_1 \) be a chord of the upper base, parallel to \( AB \). The plane passing through the lines \( AB \) and \( A_1B_1 \) forms an acute angle \( \alpha \) with the plane of the lower base of the cylinder, and the line \( AB_1 \) forms an angle \( \beta \) with the same plane. Find the height of the cylinder if the radius of the base of the cylinder is \( R \) (the points \( A \) and \( A_1 \) lie on the same side of the line passing through the midpoints of the segments \( AB \) and \( A_1B_1 \)).

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