
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 hyperbola $N$ and the four intersection points of ellipse $M$ form the vertices of a regular hexagon, along with the two foci of ellipse $M$, find the eccentricity of ellipse $M$ and the eccentricity of hyperbola $N$.

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