
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 hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$, let $M$ and $N$ be two points on the hyperbola symmetric about the origin, and let $P$ be a moving point on the hyperbola. The slopes of lines $PM$ and $PN$ are $k_{1}$ and $k_{2}$ respectively $(k_{1} \cdot k_{2} \neq 0)$. If the minimum value of $|k_{1}| + |k_{2}|$ is $1$, find the eccentricity of the hyperbola.

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