
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 $\left\{\begin{array}{l}{3x+2y=m+2}\\{2x+y=m-1}\end{array}\right.$, where $x$ and $y$ are both positive numbers.<br/>$(1)$ Find the values of $x$ and $y$ (expressed in terms of $m$);<br/>$(2)$ Determine the range of $m$; <br/>$(3)$ When $m$ is a positive integer, find the maximum value of $s=2x-3y+m$.

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