
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 linear equations in two variables $x$ and $y$: $\left\{{\begin{array}{l}{x+2y=2m+1}\\{2x+y=m+2}\end{array}}\right.$, if the solution satisfies $x-y \gt 2$, then the largest integer value of $m$ is $m=$____.

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