
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 $A$ and $B$ are the two endpoints of the major axis of the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, and $M$ and $N$ are two points on the ellipse symmetric about the $x$-axis. The slopes of the lines $AM$ and $BN$ are $k_{1}$ and $k_{2}$, respectively, with $k_{1}k_{2} \neq 0$. If the minimum value of $|k_{1}|+|k_{2}|$ is $1$, find the eccentricity of the ellipse.

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