
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 sphere $O$, with $AH:HB=1:2$, and $AB \perp$ plane $\alpha$ with $H$ being the perpendicular foot, the cross-sectional area of the sphere $O$ cut by plane $\alpha$ is $\pi$. Find the volume of sphere $O$.

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