KöMaL problem A, 2024

Let $ABCD$ be a convex cyclic quadrilateral satisfying $AB\cdot CD=AD\cdot BC$. Let the inscribed circle $\omega$ of triangle $ABC$ be tangent to sides $BC$, $CA$ and $AB$ at points $A'$, $B'$ and $C'$, respectively. Let point $K$ be the intersection of line $ID$ and the nine-point-circle of triangle $A'B'C'$ that is inside line segment $ID$. Let $S$ denote the centroid of triangle $A'B'C'$. Prove that lines $SK$ and $BB'$ intersect each other on circle $\omega$.