
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
Given that H is a point on the diameter AB of a sphere with center O, with the ratio AH:HB = 1:3, and AB is perpendicular to a plane α. Point H is the foot of the perpendicular from AB to plane α, and the cross-sectional area of the sphere O obtained by the intersection with plane α is π. Find the radius of sphere O.

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