SHARYGIN 2019 Problem 23

In the plane, let $a$ , $b$ be two closed broken lines (possibly self-intersecting), and $K$ , $L$ , $M$ , $N$ be four points. The vertices of $a$ , $b$ and the points $K$ $L$ , $M$ , $N$ are in general position (i.e. no three of these points are collinear, and no three segments between them concur at an interior point). Each of segments $KL$ and $MN$ meets $a$ at an even number of points, and each of segments $LM$ and $NK$ meets $a$ at an odd number of points. Conversely, each of segments $KL$ and $MN$ meets $b$ at an odd number of points, and each of segments $LM$ and $NK$ meets $b$ at an even number of points. Prove that $a$ and $b$ intersect.