
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 a system of one-variable linear inequalities about $x$: $\left\{\begin{array}{l}{\frac{x-1}{3}≤1}\\{2(x-1)＞m}\end{array}\right.$ has solutions and at most $3$ integer solutions. Also, the solutions of the fractional equation about $y$: $\frac{y}{2y-4}-1=\frac{m-4}{y-2}$ are non-negative integers. Find the sum of all integer values of $m$ that satisfy the conditions.

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