OBM 2013 Problem 6

The incircle of triangle $ABC$ touches sides $BC, CA$ and $AB$ at points $D, E$ and $F$ , respectively. Let $P$ be the intersection of lines $AD$ and $BE$ . The reflections of $P$ with respect to $EF, FD$ and $DE$ are $X,Y$ and $Z$ , respectively. Prove that lines $AX, BY$ and $CZ$ are concurrent at a point on line $IO$ , where $I$ and $O$ are the incenter and circumcenter of triangle $ABC$ .