If the solution set for the system of inequalities concerning x is x > 4, and an integer m allows the system of linear equations concerning x and y (where both x and y are integers) to have integer solutions, what is the sum of all integers m that satisfy the conditions?

Single Choice

If the solution set for the system of inequalities $\begin{cases} \frac{x-m}{2} > 0\\ \frac{x-4}{3} - x < -4 \end{cases}$ concerning x is x > 4, and an integer m allows the system of linear equations $\begin{cases} mx+y=8\\ 3x+y=1 \end{cases}$ concerning x and y (where both x and y are integers) to have integer solutions, what is the sum of all integers m that satisfy the conditions?