
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 blue paint and released between two perfectly smooth concentric spheres with radii of 4 cm and 6 cm respectively (the ball was outside the smaller sphere but inside the larger one). Upon contact with both spheres, the ball leaves a blue mark. During its motion, the ball traced a closed path, resulting in a region on the smaller sphere being bounded by a blue contour with an area of 27 square cm. Determine the area of the region bounded by the blue contour on the larger sphere. Give your answer in square centimeters, rounded to the nearest hundredth if necessary.

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