
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 $\frac{x^{2}}{m+1}+y^{2}=1$ $(m > 0)$ with two foci $F_{1}$ and $F_{2}$, and $E$ is a common point of the line $y=x+2$ and the ellipse, calculate the eccentricity of the ellipse when the sum of distances $|EF_{1}|+|EF_{2}|$ reaches its minimum value.

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