Number of solutions to a Diophantine Equation II

Consider the following Diophantine Equation : $x^2+y^2+z^2-xy-yz-zx-x-y-z=0$ If x, y, z are all integers, then it is a solution to the Diophantine Equation. For example, $x=4,y=6,z=9$ is one solution to the above Diophantine Equation: $4^2+6^2+9^2-4\times 6-6\times 9-9\times 4-4-6-9=0$ How many solutions are there to the above Diophantine Equation for $0<x\le y\le z\le {10}^{14}$? What is the sum of all these solutions? Answer format: count,($\sum x+y+z$) mod 10⁹

You are given: 12082,178899731 for a threshold of 10000