
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 $\frac{1}{m} + \frac{2}{n} = 1 (m > 0, n > 0)$, find the eccentricity of the ellipse $\frac{x^2}{m^2} + \frac{y^2}{n^2} = 1$ when $mn$ takes the minimum value.

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