Lattice Points on a Circle
Let $f(N)$ be the number of points with integer coordinates that are on a circle passing through $(0,0)$, $(N,0)$,$(0,N)$, and $(N,N)$.
It can be shown that $f(10000) = 36$.
What is the sum of all positive integers $N \le 10^{11}$ such that $f(N) = 420$?