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 $N \leq 10^{11}$ such that $f (N) = 420$ ?

Write-up coming later

The complete problem is available here. An approach, code, and answer will be added later.