Number of solutions to a Diophantine Equation I

Consider the following Diophantine Equation : $x^2-y^2-z^2-xy-yz-zx=0$ We call an integer triple(x,y,z) a primitive solution if it is a solution to the Diophantine Equation with gcd(x,y,z)=1 For example, (3,1,1) is a primitive solution to the above Diophantine Equation: $3^2-1^2-1^2-3\times 1-1\times 1-1\times 3=0$ In fact, there are only five primitive solutions for $0<z\le y\le x\le 100$, namely (3,1,1),(19,11,1),(37,19,5),(61,29,11), and (91,41,19). The sum of these five primitive solutions is 349, i.e. $\sum x+y+z$ How many primitive solutions are there to the above Diophantine Equation for $0<z\le y\le x\le {10}^{13}$? What is the sum of all these primitive solutions? Answer format: count,($\sum x+y+z$) mod 10⁹

You are given: 479,3971343 for a threshold of 10000