MD-MO 2007 Problem 3

$ABCDA_{1}B_{1}C_{1}D_{1}$ is a cube with side length $4a$ . Points $E$ and $F$ are taken on $(AA_{1})$ and $(BB_{1})$ such that $AE=B_{1}F=a$ . $G$ and $H$ are midpoints of $(A_{1}B_{1})$ and $(C_{1}D_{1})$ , respectively. Find the minimum value of the $CP+PQ$ , where $P\in[GH]$ and $Q\in[EF]$ .