
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
If there are exactly $3$ integer solutions for the inequality system about $x$: $\left\{\begin{array}{c}6x-5≥m\\ \frac{x}{2}-\frac{x-1}{3}＜1\end{array}\right.$, and the solution to the equation about $y$: $\frac{y-2}{3}=\frac{m-2}{3}+1$ is a non-negative number, find the sum of all integers $m$ that satisfy the conditions.

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