
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
If the solution of a linear equation is also the solution of a system of linear inequalities, then the linear equation is called the associated equation of the inequality system. For example, the solution of the equation $2x-6=0$ is $x=3$, and the solution set of the inequality system $\left\{\begin{array}{l}x-2>0\\ x<5\end{array}\right.$ is $2 < x < 5$. Since $2 < 3 < 5$, the equation $2x-6=0$ is the associated equation of the inequality system $\left\{\begin{array}{l}x-2>0\\ x<5\end{array}\right.$.<br/>$(1)$ In the equations, $5x-2=0$, $\frac{3}{4}x+1=0$, $x-\left(3x+1\right)=-5$, the associated equation of the inequality system $\left\{\begin{array}{l}2x-5>3x-8\\-4x+3<x-4\end{array}\right.$ is ______$;(fill in the number)$<br/>$(2)$ If an integer is a root of an associated equation of the inequality system $\left\{\begin{array}{l}x-\frac{1}{4}<1\\ 4+2x>-7x+5\end{array}\right.$, then this associated equation can be ______$;(write one)$<br/>$(3)$ If the equations $2x-1=x+2$ and $3+x=2(x+\frac{1}{2})$ are both associated equations of the inequality system $\left\{\begin{array}{l}x<2x-m\\ x-2\leq m\end{array}\right.$, find the range of values for $m$.

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