若实数$m$使关于$x$的不等式组$\left\{\begin{array}{l}{\frac{2-x}{3}≤2+x}\\{x＜\frac{m}{3}}\end{array}\right.$有解且至多有$3$个整数解,且使关于$x$的分式方程$\frac{mx-2}{x-1}+\frac{3}{1-x}=2$有整数解,则满足条件的整数$m$有（ ）个