SHARYGIN 2018 Problem 14

Let $ABC$ be a right-angled triangle with $\angle C = 90^{\circ}$ , $K$ , $L$ , $M$ be the midpoints of sides $AB$ , $BC$ , $CA$ respectively, and $N$ be a point of side $AB$ . The line $CN$ meets $KM$ and $KL$ at points $P$ and $Q$ respectively. Points $S$ , $T$ lying on $AC$ and $BC$ respectively are such that $APQS$ and $BPQT$ are cyclic quadrilaterals. Prove that

a) if $CN$ is a bisector, then $CN$ , $ML$ and $ST$ concur;

b) if $CN$ is an altitude, then $ST$ bisects $ML$ .