
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 inequality system $\left\{\begin{array}{l}\frac{{x-4}}{3}＜x-4\\ \frac{{m-x}}{5}＜0\end{array}\right.$ with respect to $x$ is $x \gt 4$, and the fractional equation with respect to $x$ $\frac{6}{{x-3}}+1=\frac{{mx-3}}{{x-3}}$ has a positive integer solution, find the sum of all integers $m$ that satisfy the conditions.

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