6. $\frac{2}{3}$.

Analysis: Let $E$ be the midpoint of $A B$, and establish a rectangular coordinate system with $O$ as the origin, the directions of $O E, O N$, and perpendicular to the plane $A B M N$ upwards as the positive $x, y, z$ axes, respectively. Given that the dihedral angle is 60 degrees, we have $C(1,1, \sqrt{3}), D(1,-1, \sqrt{3}), M(0,-1,0)$. There exists a point $P$, and $M P=\frac{2}{3}$.
6. As shown in the figure, it is known that $A B C D$ is a rectangle, $M, N$ are the midpoints of sides $A D, B C$ respectively, $M N$ intersects $A C$ at point $O$. Fold the rectangle $M N C D$ along $M N$. Given $A B=2, B C=4$, the dihedral angle $B-M N-C$ is 60 degrees, and $P$ is a point on line segment $M D$. If the sine of the angle formed by line $A P$ and plane $A O C$ is $\frac{\sqrt{14}}{7}$, then the length of line segment $M P$ is $\qquad$