
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]


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The vertices of an equilateral triangle ABC with side length $2\sqrt{2}$ lie on the surface of a sphere with volume $4\sqrt{3}\pi$. Find the maximum distance from a point on the sphere to plane ABC.

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