6. As shown in the figure, in the regular triangular pyramid $A_{1} B_{1} C_{1}-A B C$, $A B=2, A_{1} A=2 \sqrt{3}, D, F$ are the midpoints of edges $A B, A A_{1}$ respectively, and $E$ is a moving point on edge $A C$. Then the minimum value of the perimeter of $\triangle D E F$ is $\qquad$ .
Given $\sqrt{7}+2$.
Analysis From the regular triangular prism $A_{1} B_{1} C_{1}-A B C$, we can get $A A_{1} \perp$ the base $A B C$, so $A A_{1} \perp A B, A A_{1} \perp A C$.
In the right triangle $\triangle A D F$, $D F=\sqrt{(\sqrt{3})^{2}+1^{2}}=2$, as shown in Figures 1 and 2. When the base $A B C$ and the side $A C C_{1} A_{1}$ are unfolded in the same plane, $D E+E F$ achieves its minimum value only when points $D, E, F$ are collinear.
In $\triangle A D E$, $\angle D A E=60^{\circ}+90^{\circ}=150^{\circ}$. By the cosine rule, we have:
$$
D E=\sqrt{(\sqrt{3})^{2}+1^{2}-2 \sqrt{3} \times \cos 150^{\circ}}=\sqrt{7}
$$

Therefore, the minimum perimeter of $\triangle D E F$ is $\sqrt{7}+2$.