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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 $S(10^9)$.
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