
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 segment \( AB = 8 \) intersects the plane \( \alpha \) at an angle of \( 30^\circ \) and is divided by this plane in the ratio \( 1:3 \). Find the radius of the sphere passing through points \( A \) and \( B \) and intersecting the plane \( \alpha \) in a circle of the smallest radius.

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