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, and the endpoints of each pair of chords can move on the surface of the sphere. Given 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, the true propositions are ( ).
(A) (1)(3)(4)
(B) (1)(2)(3)
(C) (1)(2)(4)
(D) (2)(3)(4)
6. A.

Assume $AB$ and $CD$ intersect at point $N$, then $AB$ and $CD$ are coplanar, so points $A, B, C, D$ are concyclic. Clearly, in this circle, the length of chord $AB$ passing through point $N$ is greater than or equal to the length of chord $CD$ with $N$ as its midpoint, so proposition (2) is incorrect.

It is easy to prove: when the circular plane with $AB$ as its diameter is parallel to the circular plane with $CD$ as its diameter and they are on opposite sides of the sphere's center, $MN$ is maximized at 5, hence proposition (3) is correct; when the circular plane with $AB$ as its diameter is parallel to the circular plane with $CD$ as its diameter and they are on the same side of the sphere's center, $MN$ is minimized at 1, hence proposition (4) is correct.
Proposition (1) is clearly correct.