
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 is dipped in red paint and placed between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the ball is outside the smaller sphere but inside the larger sphere). As the ball moves and touches both spheres, it leaves a red trace. During its motion along a closed path, the ball creates a red-bordered area on the smaller sphere, with an area of 37 square centimeters. Find the area of the region bordered in red on the larger sphere. Provide the answer in square centimeters, rounded to two decimal places if necessary.

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