83. The length of the edge of the cube $A B C D A_{1} B_{1} C_{1} D_{1}$ is $a$. Points $P, K, L$ are the midpoints of the edges $A A_{1}, A_{1} D_{1}, B_{1} C_{1}$ respectively, and point $Q$ is the center of the face $C C_{1} D_{1} D$. The segment $M N$ has endpoints on the lines $A D$ and $K L$ and intersects the line $P Q$ perpendicularly. Find the length of this segment.
83. Note that the segment $M N$ is bisected by its intersection with the line $P Q$. Let's project this segment onto the plane $A B C D$. If $N_{1}$ is the projection of $N$, $K_{1}$ is the midpoint of $A D$, and $Q_{1}$ is the midpoint of $D C$ (where $K_{1}$ and $Q_{1}$ are the projections of $K$ and $Q$), then $N_{1} M$ is perpendicular to $A Q_{1}$ and is bisected by the point of intersection. Therefore, $\widehat{N_{1} A D}=2 \widehat{Q_{1} A D}$. From this, we can find $\left|N_{1} K_{1}\right|$, and then $\left|N_{1} M\right|$.

Answer: $\frac{a}{3} \sqrt{14}$.