
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
An ellipse has a circle passing through its two foci, with the circle's diameter endpoints being the foci. The circle and ellipse intersect at four distinct points. When these four points and the two foci are connected in sequence, they form a regular hexagon. Determine the eccentricity of this ellipse.

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