
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
A sphere with a radius of 2 contains an inscribed right square prism with a base edge length of 2. Determine the difference between the surface area of the sphere and the lateral area of the prism when the lateral area of the prism is at its maximum.

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