CMO 2016 Problem 2

In $\triangle AEF$ , let $B$ and $D$ be on segments $AE$ and $AF$ respectively, and let $ED$ and $FB$ intersect at $C$ . Define $K,L,M,N$ on segments $AB,BC,CD,DA$ such that $\frac{AK}{KB}=\frac{AD}{BC}$ and its cyclic equivalents. Let the incircle of $\triangle AEF$ touch $AE,AF$ at $S,T$ respectively; let the incircle of $\triangle CEF$ touch $CE,CF$ at $U,V$ respectively. Prove that $K,L,M,N$ concyclic implies $S,T,U,V$ concyclic.