
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 $\frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1$ ($a>b>0$) and a hyperbola $\frac {x^{2}}{m^{2}}- \frac {y^{2}}{n^{2}}=1$ ($m>0$, $n>0$) 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}
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