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

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