
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 equation of the ellipse $C$ is $\frac{{x}^{2}}{{a}^{2}}+\frac{{{y}^{2}}}{{{b}^{2}}}=1(a > b > 0)$, one asymptote of the hyperbola $\frac{{{x}^{2}}}{{{a}^{2}}}- \frac{{{y}^{2}}}{{{b}^{2}}}=1$ makes an angle of $30^{\circ}$ with the $x$-axis, and the focal length of the hyperbola is $4\sqrt{2}$.

(1) Find the equation of the ellipse $C$;
(2) Let $F\_{1}$, $F\_{2}$ be the left and right foci of the ellipse $C$, respectively. Draw a line $l$ passing through $F\_{2}$ and intersecting the ellipse at points $A$ and $B$ ($l$ is not parallel to the $x$-axis). Let $E$ be the midpoint of segment $AB$. Find the range of the slope $k$ of the line passing through $F\_{1}$ and $E$.

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