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 $-X\le a<b<c\le X$.

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