4. Given a regular quadrilateral pyramid $S-A B C D$ with side length of the lateral edges being 4, and $\angle A S B=30^{\circ}$. Points $E$, $F$, and $G$ are taken on the lateral edges $S B$, $S C$, and $S D$ respectively. The minimum value of $A E+E F+F G+G A$ is $\qquad$
Therefore, the minimum value of $A E+E F+F G+G A$ is
$$
2 S A \sin 60^{\circ}=4 \sqrt{3} \text {. }
$$

Hint: Unfold the lateral faces of the pyramid along $S A$, as shown in Figure 14.