Three, the edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ is 1. On the face diagonal $A_{1} D$, take point $M$, and on $C D_{1}$, take point $N$, such that the line segment $M N$ is parallel to the diagonal plane $A_{1} A C C_{1}$. Calculate the minimum length of the line segment $M N$ that satisfies the above conditions.