
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 the solution set of the system of linear inequalities in one variable $x$, $\left\{{\begin{array}{l}{\frac{{x-m}}{6}≥0,}\\{x+3＜3(x-1)}\end{array}}\right.$, is $x \gt 3$, and the fractional equation in variable $y$, $\frac{{3-y}}{{2-y}}=3-\frac{m}{{y-2}}$, has a non-negative integer solution, find the sum of the values of the integer $m$ that satisfy the conditions.

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