
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
Given the system of equations in terms of $x$ and $y$: $\left\{\begin{array}{l}nx+(n+1)y=n+2\\ x-2y+mx=-5\end{array}\right.$ ($n$ is a constant).<br/>$(1)$ When $n=1$, the system of equations can be simplified to $\left\{\begin{array}{l}x+2y=3\\ x-2y+mx=-5\end{array}\right.$.<br/>① Please directly write down all non-negative integer solutions of the equation $x+2y=3$.<br/>② If the solutions of this system also satisfy the equation $x+y=2$, find the value of $m$.<br/>$(2)$ When $n=3$, if the system of equations has integer solutions, find the integer value of $m$.

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