
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
A new definition: If the solution of a linear equation is within the solution set of a system of linear inequalities, then the linear equation is called the "associated equation" of the system of inequalities. For example, the solution of the equation $x-1=3$ is $x=4$, and the solution set of the system of inequalities $\left\{\begin{array}{l}{x-1>1}\\{x-2<3}\end{array}\right.$ is $2 < x < 5$. It is easy to see that $x=4$ is within the range of $2 < x < 5$, so the equation $x-1=3$ is the "associated equation" of the system of inequalities $\left\{\begin{array}{l}{x-1>1}\\{x-2<3}\end{array}\right.$.

$(1)$ In the equations $2(x+1)-x=-3$; $\frac{x+1}{3}-1=x$; $2x-7=0$, the "associated equation" of the system of inequalities $\left\{\begin{array}{l}{2x+1>x-3}\\{3(x-2)-x\leq 2}\end{array}\right.$ is ______$; (fill in the numbers)$<br/>$(2)$ The equation in terms of $x$, $2x-k=6$, is the "associated equation" of the system of inequalities $\left\{\begin{array}{l}{\frac{x-3}{2}\geq x}\\{\frac{2x+5}{2}>\frac{1}{2}x}\end{array}\right.$, find the range of values for $k$;<br/>$(3)$ If the equation in terms of $x$, $\frac{x+5}{2}-3m=0$, is the "associated equation" of the system of inequalities $\left\{\begin{array}{l}{\frac{x+2m}{2}>m}\\{x-m\leq 2m+1}\end{array}\right.$, and at this time the system of inequalities has $3$ integer solutions, try to find the range of values for $m$.

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