SHARYGIN 2021 Problem 7

The incircle of triangle $ABC$ centered at $I$ touches $CA,AB$ at points $E,F$ respectively. Let points $M,N$ of line $EF$ be such that $CM=CE$ and $BN=BF$ . Lines $BM$ and $CN$ meet at point $P$ . Prove that $PI$ bisects segment $MN$ .