
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 $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ $(a>0, b>0)$, the distance from the focus to the asymptote is $\sqrt{3}$, and the eccentricity is $\frac{\sqrt{7}}{2}$. <br/>$(1)$ Find the equation of the hyperbola $C$; <br/>$(2)$ The line $l: y=kx+3$ intersects the hyperbola at points $M$ and $N$, and if $|MN|=16\sqrt{3}$, find the value of $k$.

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