BALKAN-SL 2019 G8

Given an acute triangle $A B C,(c)$ is circumcircle with center $O$ and $H$ the orthocenter of the triangle $A B C$. The line $A O$ intersects $(c)$ at the point $D$. Let $D_{1}, D_{2}$ and $H_{2}, H_{3}$ be the symmetrical points of the points $D$ and $H$ with respect to the lines $A B, A C$ respectively. Let $\left(c_{1}\right)$ be the circumcircle of the triangle $A D_{1} D_{2}$. Suppose that the line $A H$ intersects again $\left(c_{1}\right)$ at the point $U$, the line $H_{2} H_{3}$ intersects the segment $D_{1} D_{2}$ at the point $K_{1}$ and the line $D H_{3}$ intersects the segment $U D_{2}$ at the point $L_{1}$. Prove that one of the intersection points of the circumcircles of the triangles $D_{1} K_{1} H_{2}$ and $U D L_{1}$ lies on the line $K_{1} L_{1}$.