
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
Definition: If the solution of a linear equation is also the solution of a system of linear inequalities, then the linear equation is called the "companion equation" of the inequality system. For example, the solution of the equation $2x-6=0$ is $x=3$. The solution set of the inequality system $\left\{\begin{array}{l}x-2>0\\ x<5\end{array}\right.$ is $2 < x < 5$ because $2 < 3 < 5$. Therefore, the equation $2x-6=0$ is called the "companion equation" of the inequality system $\left\{\begin{array}{l}x-2>0\\ x<5\end{array}\right.$.

$(1)$ Among the following equations, the "companion equation" of the inequality system $\left\{\begin{array}{l}x+1>0\\ x<2\end{array}\right.$ is ______$; (fill in the number)$<br/>①$x-1=0$; ②$2x+1=0$; ③$-2x-2=0$<br/>$(2)$ If the equation about $x$, $2x-k=2$, is the "companion equation" of the inequality system $\left\{{\begin{array}{l}{3x-6>4-x}\\{x-1\geq 4x-10}\end{array}}\right.$, find the range of $k$;<br/>$(3)$ If the equations $2x+4=0$ and $\frac{{2x-1}}{3}=-1$ are both the "companion equations" of the system of inequalities $\left\{\begin{array}{l}({m-2})x<m-2\\ x+5\geq m\end{array}\right.$, where $m > 2$, then the range of $m$ is ______$(write the answer directly)$.

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