
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 center of the hyperbola is at the origin and one focus is at $F\left(2,0\right)$, the line $y=x-1$ intersects the hyperbola at points $A$ and $B$, with the midpoint of $AB$ having a horizontal coordinate of $-\frac{1}{2}$.  
$(1)$ Find the equation of the hyperbola.  
$(2)$ Let $A_{1}$ and $A_{2}$ be the two endpoints of the real axis of the hyperbola. If the line $l$ passing through $F$ intersects the hyperbola $C$ at points $M$ and $N$, investigate whether the intersection point $Q$ of the lines $A_{1}M$ and $A_{2}N$ lies on a certain fixed line. If so, find the equation of that fixed line; if not, explain the reason.

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