6 The line segment connecting two points on a sphere is called a chord of the sphere. For a sphere with a radius of 4, the lengths of two chords $AB$ and $CD$ are $2\sqrt{7}$ and $4\sqrt{3}$, respectively. $M$ and $N$ are the midpoints of $AB$ and $CD$, respectively. The endpoints of the two chords can move on the surface of the sphere. There are the following four propositions:
(1) Chords $AB$ and $CD$ may intersect at point $M$;
(2) Chords $AB$ and $CD$ may intersect at point $N$;
(3) The maximum value of $MN$ is 5;
(4) The minimum value of $MN$ is 1.
Among these propositions, the true ones are ( ).
(A) (1)(3)(4)
(B) (1)(2)(3)
(C) (1)(2)(4)
(D) (2)(3)(4)
6 Assume $A B$ and $C D$ intersect at point $N$, then $A B$ and $C D$ are coplanar, so points $A, B, C, D$ are concyclic. The length of chord $A B$ through the midpoint $N$ of chord $C D$ is clearly $A B \geqslant C D$, so (2) is incorrect. It is easy to prove that when the circular plane with diameter $A B$ is parallel to the circular plane with diameter $C D$ and they are on opposite sides of the sphere's center, $M N$ is maximized at 5, hence (3) is correct. When the circular plane with diameter $A B$ is parallel to the circular plane with diameter $C D$ and they are on the same side of the sphere's center, $M N$ is minimized at 1, hence (4) is correct. Therefore, the answer is $\mathrm{A}$.