MathNet 2014 Question 41

Circles $w_1, w_2$ with equal radius $R$ intersect in two points. The line joining centres of these circles intersects $w_1$ in points $A$, $C$ and intersects $w_2$ in points $B$, $D$. ($B$ is between $A$ and $C$, $C$ is between $B$ and $D$). Draw a circle with diameter $AD$ tangent internally $w_1$ at $A$ and tangent internally to $w_2$ at $D$. Let $l_1$ be tangent line to the $w_1$ at point $C$ and let $l_2$ be tangent line to the $w_2$ at point $B$. $l_2$ intersects $w_3$ in a point $K$. Tangent lines from the point $K$ to the circle $w_1$ intersects line $l_1$ in $P$ and $Q$. Find length of line segment $PQ$.
(proposed by G. Munkhbayar)