
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 a hyperbola $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ ($a>0, b>0$) with eccentricity $2$, and the hyperbola $C$ passes through the point $P\left(\frac{\sqrt{6}}{2}, -\frac{\sqrt{6}}{2}\right)$.  
$(1)$ Find the equation of the hyperbola $C$;  
$(2)$ Let $M$ be any point on the line $x=\frac{1}{2}$. Two tangent lines $l_{1}$ and $l_{2}$ are drawn through point $M$ on the hyperbola $C$, with tangent points $A$ and $B$ respectively. Determine whether the line $AB$ passes through a fixed point. If it passes through a fixed point, find the coordinates of that point; if it does not pass through a fixed point, explain the reason.

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