
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 line $l$ that does not coincide with the $x$-axis passes through the point $N(x_{N}, 0)$ where $x_{N} \neq 0$, and the hyperbola $C: \frac{{x}^{2}}{{a}^{2}} - \frac{{y}^{2}}{{b}^{2}} = 1$ where $a > 0$ and $b > 0$, with points $A$ and $B$ symmetric with respect to line $l$, and the $x$-coordinate of the midpoint $M$ of $AB$ is $x_{M}$, if $x_{N} = 4x_{M}$, calculate the eccentricity of $C$.

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