
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
New definition: If the roots of a linear equation are the solutions of a system of linear inequalities, then the linear equation is called the associated equation of the inequality system.  
$(1)$ In the equations $①$ $2x-1=0$; $②$ $2x+1=0$; $③$ $x-\left(3x+1\right)=-5$, the associated equation of the inequality system $\left\{\begin{array}{l}{-x+2>x-5}\\{3x-1>-x+2}\end{array}\right.$ is ______ (fill in the number).  
$(2)$ If an integer is a root of a certain associated equation of the inequality system $\left\{\begin{array}{l}{x-2<1}\\{1+x>-x+2}\end{array}\right.$, then this associated equation can be _______. (Write down one equation)  
$(3)$ If the equations $1-x=x$, $3+x=2({x+\frac{1}{2}})$ are both associated equations of the inequality system $\left\{\begin{array}{l}{x<2x-m}\\{x-2\leq m}\end{array}\right.$, directly write down the range of values for $m$.

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