
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 $y$: $\left\{\begin{array}{l}2x+y=1+2m\\ x+2y=2-m\end{array}\right.$, where the solution satisfies the inequality $x+y \gt 0$. Find the range of real numbers for $m$. If the inequality $\left(2m+1\right)x-2m \lt 1$ has a solution $x \gt 1$ under the condition in (1), determine the integer values of $m$.

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