
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 an ellipse $M: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, and a hyperbola $N: \frac{x^2}{m^2} - \frac{y^2}{n^2} = 1$. If the two asymptotes of the hyperbola $N$ intersect with the ellipse $M$ at four points, and these four points along with the two foci of the ellipse $M$ exactly form the vertices of a regular hexagon, then the eccentricity of the ellipse $M$ is __________.

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