
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 a sphere $O$ whose center is at a distance equal to half the radius of the sphere from a plane passing through three points $A$, $B$, and $C$ on the surface of the sphere, and given $AB=3$, $\tan \angle ACB= -\sqrt{3}$, then the volume of sphere $O$ is _______.

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