6 Given that the edge length of the regular tetrahedron $ABCD$ is 2, the sum of the areas of the sections obtained by all planes equidistant from its four vertices is $\qquad$ .
(6) $3+\sqrt{3}$ Hint: Such cross-sections fall into two categories:
(1) One side of the cross-section has one vertex, and the other side has three vertices, such as $\triangle B_{1} C_{1} D_{1}$ in Figure (1), where $B_{1} 、 C_{1} 、 D_{1}$ are the midpoints of edges $A B 、 A C 、 A D$, respectively. There are four such cross-sections, and the area of each is $\frac{\sqrt{3}}{4}$;
(Problem 6)
(2) Each side of the cross-section has two vertices, such as quadrilateral $M N P Q$ in Figure (2), where $M 、 N 、 P 、 Q$ are the midpoints of edges $A B 、 B C 、 C D 、 A D$, respectively. There are three such cross-sections, all of which are squares, and the area of each is 1. In summary, the total area of all cross-sections is $3+\sqrt{3}$.