
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 right prism $ABCD-A_{1}B_{1}C_{1}D_{1}$ where each vertex lies on the surface of a sphere $O$, if the surface area of the sphere $O$ is $12\pi$, then the maximum value of the lateral surface area of the prism is ____.

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