Value of distances between randomly generated points in a 3D projection

Question

Suppose you Randomly generate n points using a uniform distribution, (x,y,z) where x,y, and z are between 0 and 1. Calculate the distance between a point and all the other points calculated, then take the average of those distances and assign that value to that point. Repeat the process for all points in the set, then take the average of all generated values and label it k. As n approaches infinity, what does the value of k converge to?