
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 $C$: $\frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ ($a>0, b>0$) with the right focus $F$ and left vertex $A$, where $|FA|=2+\sqrt{5}$, the distance from $F$ to the asymptote of $C$ is $1$. A line $l$ passing through point $B(4,0)$ intersects the right branch of the hyperbola $C$ at points $P$ and $Q$. The lines $AP$ and $AQ$ intersect the $y$-axis at points $M$ and $N$, respectively. <br/>$(1)$ Find the standard equation of the hyperbola $C$;<br/>$(2)$ If the slopes of lines $MB$ and $NB$ are $k_{1}$ and $k_{2}$, respectively, determine whether $k_{1}k_{2}$ is a constant. If it is, find the value of this constant; if not, explain why.

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