
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
For the partial solutions of the binary linear equation in $x$ and $y$, $ax+by=c$, as shown in the table:<br/>

| $x$ | $\ldots $ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $\ldots $ |
|-----|-----------|------|------|----|----|----|----------|
| $y$ | $\ldots $ | $-1$ | $0$  | $1$| $2$| $3$| $\ldots $ |

$(1)$ The binary linear equation is ________;<br/>
$(2)$ If the solution of the system of equations in $x$ and $y$, $\left\{\begin{array}{l}{ax+by=c}\\{2x-y=-m}\end{array}\right.$, is positive, find the range of $m$;<br/>
$(3)$ When $x \gt 3$, for each value of $x$, let the value of $y$ in the equation $ax+by=c$ be denoted as $y_{1}$, and the value of $y$ in the equation $2x-y=-m$ be denoted as $y_{2}$. If $y_{1} \lt y_{2}$, find the range of $m$.

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