
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 ball with a radius of 1 cm was dipped in red paint and placed between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the ball is outside the smaller sphere but inside the larger one). As it moves and touches both spheres, the ball leaves a red mark. During its movement, the ball traveled along a closed path, resulting in a red-bordered area on the smaller sphere with an area of 37 square centimeters. Find the area of the region bordered by the red mark on the larger sphere. Give your answer in square centimeters, rounding to hundredths if necessary.

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