
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 that the hyperbola $C$ shares asymptotes with the hyperbola $\frac{{x}^{2}}{4}-\frac{{y}^{2}}{2}=1$, and passes through the point $(-3,\frac{\sqrt{6}}{2})$.  
$(1)$ Find the standard equation of the hyperbola $C$;  
$(2)$ Let $P$ be any point on the line $x=2$. Two tangent lines $PA$ and $PB$ are drawn from point $P$ to the hyperbola $C$, with tangents at points $A$ and $B$ respectively. Lines perpendicular to the $x$-axis are drawn from the right vertex of hyperbola $C$ to lines $PA$ and $PB$ at points $M$ and $N$ respectively. The vertical coordinates of points $M$ and $N$ are $m$ and $n$ respectively. Find the value of $mn$.

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