
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 placed between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the ball ended up outside the smaller sphere but inside the larger one). When touching both spheres, the ball leaves a blue mark. During its motion, the ball traveled along a closed route, resulting in a blue-bordered region on the smaller sphere with an area of 27 square cm. Find the area of the region bordered in blue on the larger sphere. Provide your answer in square centimeters, rounded to two decimal places if necessary.

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