
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 that the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) and the hyperbola $\frac{x^2}{m^2} - \frac{y^2}{n^2} = 1$ ($m > 0, n > 0$) have the same foci at $(-c,0)$ and $(c,0)$, and $c$ is the geometric mean of $a$ and $m$, and $n^2$ is the arithmetic mean of $2m^2$ and $c^2$, find the eccentricity of the ellipse.

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