
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 large ornament ball is designed with a spherical core and an external cylindrical decorative layer. The outer cylindrical layer fits tightly around the sphere, sharing the same vertical axis. The diameter of the sphere is 12 inches. The height of the cylinder is equal to the diameter of the sphere. Calculate the total surface area of this decorative ball, excluding the top and bottom of the cylindrical layer, but including its lateral surface. Express your answer as a common fraction in terms of \(\pi\).

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