
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
Cut a sphere with radius $R$ using two parallel planes so that the surface area of the spherical segment between them is constant, but the volume of the spherical layer is maximized.

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