
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 each with radius 1 that are externally tangent to each other and tangent to a cylindrical surface with radius 1, and a smaller sphere that is tangent to both large spheres and also tangent to the cylindrical surface, determine the maximum possible value of the eccentricity of the ellipse formed by the intersection of a plane passing through the centers of the smaller and one of the larger spheres with the cylindrical surface.

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