4. In the unit cube $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there is a point $M$ on the side face $A A^{\prime} B^{\prime} B$ such that the distances from $M$ to the lines $A B$ and $B^{\prime} C^{\prime}$ are equal. The minimum distance from a point on the trajectory of $M$ to $C^{\prime}$ is $\qquad$
The locus of point $M$ is a parabola with point $B^{\prime}$ as the focus, $A B$ as the directrix, and passing through point $A^{\prime}$. When $M$ is the midpoint $F$ of $B B^{\prime}$, the distance from point $M$ to $C^{\prime}$ is minimized, with the minimum value being $\frac{\sqrt{5}}{2}$.