
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 \( ABC A_{1} B_{1} C_{1} \) is a right triangle \( ABC \) where the larger leg \( AB \) is equal to \( a \) and the angle opposite to it \( C \) is \( \alpha \). The hypotenuse \( BC \) is the diameter of the base of a cone whose vertex lies on the edge \( A_{1} B_{1} \). Find the height of the cone, given that \( A A_{1} = \frac{a}{2} \).

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