
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 equation of one of the asymptotes of hyperbola $C$ is $x-2y=0$, and the hyperbola $C$ passes through the point $(2\sqrt{2},1)$.

$(1)$ Find the equation of the hyperbola $C$;

$(2)$ Suppose the left and right vertices of the hyperbola $C$ are $A_{1}$ and $A_{2}$, respectively, and $P$ is any point on $C$. The lines $PA_{1}$ and $PA_{2}$ intersect the line $l:x=1$ at $M$ and $N$, respectively. Find the minimum value of $|MN|$.

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