Circles ${\omega }_1,{\omega }_2$ and ${\omega }_3$ each have radius $4$ and are placed in the plane so that each circle is externally tangent to the other two. Points $P_1,P_2,P_3$ lie on ${\omega }_1,{\omega }_2$ and ${\omega }_3$ respectively such that $P_1{P}_2=P_2{P}_3=P_3{P}_1$ and line $P_i{P}_{i+1}$ is tangent to ${\omega }_i$ for each $i=1,2,3$, where $P_4=P_1$. Let $S$ be the side length of $△P_1{P}_2{P}_3$. $S^2$ can be written in the form $X+\sqrt{Y}$ for positive integers $X$ and $Y$. What is $X+Y$?