Example 3 Given a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with edge length 1, $A C_{1}$ is a diagonal. $M, N$ are the midpoints of $B B_{1}$, $B_{1} C_{1}$ respectively, and $P$ is the midpoint of line segment $M N$. Find the distance between the skew lines $D P$ and $A C_{1}$.