
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
Given two large spheres with radius 1 that are externally tangent to each other and both inscribed within a cylinder with radius 1, and a smaller sphere that is externally tangent to both large spheres and also inscribed within the cylinder. A plane passing through the centers of the smaller sphere and one of the large spheres intersects the cylinder to form an ellipse. What is the maximum possible value of the eccentricity of this ellipse?

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