
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
In the Cartesian coordinate system, for point $P(a,b)$ and point $Q(x,y)$, the following definition is given: if $\left\{\begin{array}{l}x=a+2\\ y=b-4\end{array}\right.$, then point $Q$ is called the companion point of point $P$, for example: the companion point of point $(2,0)$ is $(4,-4)$.  
$(1)$ If the companion point $Q$ of point $P(2,-1)$ lies on the hyperbola $y=\frac{k}{x}$ (where $k\neq 0$), then the value of $k$ is ______.  
$(2)$ Given that point $P(a,b)$ lies on the line $y=x-m$, and point $Q(-1,-2)$ is the companion point of point $P$, find the value of $m$.  
$(3)$ Given that point $P(a,b)$ lies on the line $y=2x+3$, and the companion point $Q$ of point $P$ also lies on a line, find the function expression corresponding to the line where point $Q$ lies.

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