
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 a hyperbola $C:\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1(a＞0,b＞0)$, one of its asymptotes is perpendicular to the line $x+2y=0$, and the distance from the right vertex $A$ to this asymptote is $\frac{2\sqrt{5}}{5}$.<br/>$(1)$ Find the equation of the hyperbola $C$;<br/>$(2)$ If the line $l$ intersects the hyperbola $C$ at points $A$ and $B$, with the midpoint of segment $AB$ at $M\left(3,2\right)$, find the slope of the line $l$.

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