A region $S$ in the complex plane is defined by $S=x+iy:-2\le x\le 2,-2\le y\le 2$. A complex number $z=x+iy$ is chosen uniformly at random from $S$. What is the probability that $\left(\frac{2}{3}+\frac{2}{3}i\right)z$ is also in $S$?