CN-TST 2018 Problem 13

Let $\omega_1,\omega_2$ be two non-intersecting circles, with circumcenters $O_1,O_2$ respectively, and radii $r_1,r_2$ respectively where $r_1 < r_2$ . Let $AB,XY$ be the two internal common tangents of $\omega_1,\omega_2$ , where $A,X$ lie on $\omega_1$ , $B,Y$ lie on $\omega_2$ . The circle with diameter $AB$ meets $\omega_1,\omega_2$ at $P$ and $Q$ respectively. If $$\angle AO_1P+\angle BO_2Q=180^{\circ},$$ find the value of $\frac{PX}{QY}$ (in terms of $r_1,r_2$ ).