Count all the points

Algebra Level 3

Let (x,y,z)(x,y,z) be points with integer co-ordinates satisfying the system of homogeneous equations {3xyz=03x+z=03x+2y+z=0\begin{cases} 3x-y-z=0 \\ -3x+z=0 \\ -3x+2y+z=0 \end{cases} Find the number of such points for which x2+y2+z2100x^2+y^2+z^2 \leq 100


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