
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
The left vertex of the hyperbola $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1 (a>0, b>0)$ is $A$, and the focal length is $4$. A line passing through the right focus $F$ perpendicular to the real axis intersects $C$ at points $B$ and $D$, and $\triangle ABD$ is a right triangle.  
$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ Let $M$ and $N$ be two moving points on the right branch of $C$. If the slopes of the lines $AM$ and $AN$ are $k_{1}$ and $k_{2}$ respectively, and $k_{1}k_{2}=-2$, find the range of values of the distance $d$ from point $A$ to the line $MN$.

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