
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 the ellipse $M: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ and the hyperbola $N: \frac{x^{2}}{m^{2}}-\frac{y^{2}}{n^{2}}=1$, if the two asymptotes of the hyperbola $N$ and the four intersection points of the ellipse $M$ and the two foci of the ellipse $M$ form a regular hexagon, determine the eccentricity of the ellipse $M$.

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