
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
Within a closed right prism ABC-A<sub>1</sub>B<sub>1</sub>C<sub>1</sub> there is a sphere with volume V. Given AB is perpendicular to BC, AB=6, AC=10, and AA<sub>1</sub>=3, find the maximum volume of the sphere.

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