SHARYGIN 2017 Problem 19

Let cevians $AA', BB'$ and $CC'$ of triangle $ABC$ concur at point $P.$ The circumcircle of triangle $PA'B'$ meets $AC$ and $BC$ at points $M$ and $N$ respectively, and the circumcircles of triangles $PC'B'$ and $PA'C'$ meet $AC$ and $BC$ for the second time respectively at points $K$ and $L$ . The line $c$ passes through the midpoints of segments $MN$ and $KL$ . The lines $a$ and $b$ are defined similarly. Prove that $a$ , $b$ and $c$ concur.