IGO 2018 A4

Quadrilateral $ABCD$ is circumscribed around a circle. Diagonals $AC,BD$ are not perpendicular to each other. The angle bisectors of angles between these diagonals, intersect the segments $AB,BC,CD$ and $DA$ at points $K,L,M$ and $N$ . Given that $KLMN$ is cyclic, prove that so is $ABCD$ .

Proposed by Nikolai Beluhov (Bulgaria)