Ants on a sphere

Consider a solid sphere in the co-ordinate space, centered at the origin and having a great circle of circumference $4$ units. Let the points where this sphere intersects the axes be called "good" points. Two ants are placed on two randomly selected, distinct good points. Whenever one of the ants is at a good point, it randomly chooses an adjacent good point and moves to it with constant speed along the great circle joining the points. The ants make their decisions independently. They start moving simultaneously when the clock is at zero, and move continuously with constant speed. If it is known that each ant travels at the speed of $1$ unit per second, find the expected time elapsed(in seconds) before they meet. The answer is of the form $\frac{a}{b}$ when expressed as simplest fraction. Enter value of $\left\lceil a+\frac { 3 }{ 4 } b \right\rceil$ .

Liked this? Try similar problems in the set: Ants on an adventure!

Inspiration