
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 smooth sphere with a radius of 1 cm is dipped in red paint and placed between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the painted sphere is outside the smaller sphere but inside the larger one). As it moves, the sphere leaves a red mark when it touches either of the concentric spheres. After traveling through a closed route, a region enclosed by the red mark forms on the smaller sphere, with an area of 47 square centimeters. Find the area of the region enclosed by the red mark on the larger sphere. Provide the answer in square centimeters, rounded to the nearest hundredth if necessary.

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