In a spatial coordinate system, there is a globe with center at $O ( 0,0,0 )$ and north pole at $N ( 0,0,2 )$. A point $A$ on the sphere has coordinates $\left( \frac { \sqrt { 3 } } { 2 } , \frac { 1 } { 2 } , \sqrt { 3 } \right)$. The point on the equator farthest from point $A$ is point $P$. On the great circle passing through points $A$ and $P$, the length of the minor arc between these two points is (blank). (Express as a fraction in lowest terms)