RU-SPB 2015 Problem 117

Let $BL$ be angle bisector of acute triangle $ABC$ .Point $K$ choosen on $BL$ such that $\measuredangle AKC-\measuredangle ABC=90º$ .point $S$ lies on the extention of $BL$ from $L$ such that $\measuredangle ASC=90º$ .Point $T$ is diametrically opposite the point $K$ on the circumcircle of $\triangle AKC$ .Prove that $ST$ passes through midpoint of arc $ABC$ .(S. Berlov)