
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 the hyperbola $l: kx + y - \sqrt{2}k = 0$ has an asymptote parallel to an asymptote of the hyperbola $C: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ (where $a > 0$, $b > 0$), and the distance between these two parallel lines is $\frac{4}{3}$, find the eccentricity ($e$) of the hyperbola $C$.

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