
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 a subset equation of the inequalities.<br/>$(1)$ Given the following equations:<br/>① $3x-2=0$;<br/>② $2x-3=0$;<br/>③ $x-\left(3x+1\right)=-7$.<br/>Among them, the subset equations of the inequalities $\left\{\begin{array}{l}{-x+2>x-5}\\{3x-1>-x+2}\end{array}\right.$ are ______ $(fill in the numbers)$;<br/>$(2)$ Given the system of inequalities in terms of $x$: $\left\{\begin{array}{l}{x+m<2x}\\{x-2<m}\end{array}\right.$.<br/>① If the equation $\frac{2x-1}{3}=-3$ is a subset equation of this system of inequalities, find the range of $m$;<br/>② If the equations $x=1$ and $x=2$ are not subset equations of this system of inequalities, then the range of $m$ is ______.

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