HMMT 2014 G9

Two circles are said to be orthogonal if they intersect in two points, and their tangents at either point of intersection are perpendicular. Two circles $\omega_{1}$ and $\omega_{2}$ with radii 10 and 13 , respectively, are externally tangent at point $P$. Another circle $\omega_{3}$ with radius $2 \sqrt{2}$ passes through $P$ and is orthogonal to both $\omega_{1}$ and $\omega_{2}$. A fourth circle $\omega_{4}$, orthogonal to $\omega_{3}$, is externally tangent to $\omega_{1}$ and $\omega_{2}$. Compute the radius of $\omega_{4}$.