5. In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, there are points $M$, $N$, and $P$ on $C C_{1}$, $B C$, and $C D$ respectively, and $C M=C N$. To make $A_{1} P \perp$ plane $D M N$, determine the position of point $P$.
(Tip: $D P=C M$. )