
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 $\left\{\begin{array}{l}{-x-2y=1-3m,①}\\{3x+4y=2m,②}\end{array}\right.$ and the solution satisfies $x+y\geqslant 0$. Find:<br/>$(1)$ Non-negative integer solutions for $m$;<br/>$(2)$ Simplify: $|m-3|+|5-2m|$;<br/>$(3)$ For what integer values of $m$ within the range of possible values, is the solution set for the inequality $m\left(x+1\right) \gt 0$ given by $x \gt -1$.

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