BALKAN-SL 2017 G4

The acuteangled triangle $A B C$ with circumcenter $O$ is given. The midpoints of the sides $B C$, $C A$ and $A B$ are $D, E$ and $F$ respectivelly. An arbitrary point $M$ on the side $B C$, different of $D$, is choosen. The straight lines $A M$ and $E F$ intersects at the point $N$ and the straight line $O N$ cut again the circumscribed circle of the triangle $O D M$ at the point $P$. Prove that the reflection of the point $M$ with respect to the midpoint of the segment $D P$ belongs on the nine points circle of the triangle $A B C$.