若变量 $\mathit{x\hspace{0.33em}}，\mathit{\hspace{0.33em}\hspace{0.33em}y}$满足约束条件 $\left\{\begin{array}{c}2x+y+3\ge 0，\\ x-2y+4\ge 0，\\ x-2\le 0.\end{array}\right.$则 $z=x+\frac{1}{3}y$的最大值是________．