
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 line $2x-y=0$ is $2$. <br/>$(1)$ Find the standard equation of the hyperbola $C$; <br/>$(2)$ Let the left and right vertices of the hyperbola $C$ be $A$ and $B$ respectively. The line $l$ intersects the hyperbola $C$ at points $M$ and $N$ (point $M$ is in the first quadrant). Let the slope of line $MA$ be $k_{1}$ and the slope of line $NB$ be $k_{2}$. A perpendicular line is drawn from the origin $O$ to line $l$, and the foot of the perpendicular is $H$. When $\frac{k_{1}}{k_{2}}$ is a constant $-\frac{1}{3}$, is there a fixed point $G$ such that $|GH|$ is a fixed value? If yes, find this fixed point $G$. If not, explain the reason.

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