Eccoci.

On the parabola $y = x^2/k$, three points $A(a, a^2/k)$, $B(b, b^2/k)$ and $C(c, c^2/k)$ are chosen.

Let $F(K, X)$ be the number of the integer quadruplets $(k, a, b, c)$ such that at least one angle of the triangle $ABC$ is $45$-degree, with $1 \le k \le K$ and $-X \le a \lt b \lt c \le X$.

For example, $F(1, 10) = 41$ and $F(10, 100) = 12492$.
Find $F(10^6, 10^9)$.