
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
Two spheres, each with a radius of 1, are externally tangent to each other and internally tangent to a cylinder with a radius of 1. Another smaller sphere is externally tangent to both of these spheres and internally tangent to the same cylinder. If the plane passing through the center of the small sphere and the center of one of the larger spheres intersects the cylinder to form an ellipse, find the maximum possible value of the eccentricity \( e \) of this ellipse.

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