Consider the following Diophantine equation: $$15(x^2+y^2+z^2)=34(xy+yz+zx)$$ where $x$, $y$ and $z$ are positive integers.

Let $S(N)$ be the sum of all solutions, $(x,y,z)$, of this equation such that, $1\le x\le y\le z\le N$ and $gcd(x,y,z)=1$.

For $N={10}^2$, there are three such solutions - $(1,7,16),(8,9,39),(11,21,72)$. So $S({10}^2)=184$.

Find $S({10}^9)$.