Mongolian Mathematical Olympiad

A circle $\gamma$ with center $I$ and radius $R$ is inscribed in quadrilateral $ABCD$. Another circle $\omega$ with center $O$, ($I\ne O$) and radius $r$ is situated inside the quadrilateral $ABCD$. Circles ${\gamma }_B,{\gamma }_C,{\gamma }_D,{\gamma }_A$ inscribed in the angles $\angle ABC,\angle BCD,\angle CDA,\angle DAB$ are tangent to the circle $\omega$ at points $B_1,C_1,D_1,A_1$ respectively. If circles ${\gamma }_B,{\gamma }_C$ tangent to $\omega$ externally and circles ${\gamma }_D,{\gamma }_A$ tangent to $\omega$ internally and $A{A}_1\cap D{D}_1=P,B{B}_1\cap C{C}_1=Q$ then find the value of $\frac{IP+IQ}{PQ}$.