Primitive Inverse-Cubed Triplets!

Number Theory

Define a primitive inverse-cubed triplet as any ordered triplet $(x,y,z),$ where $x,y,z\in\mathbb{Z}^+$ and $\gcd{(x,y,z)}=1,$ such that

$\frac{1}{x^3}+\frac{1}{y^3}=\frac{1}{z^3}.$

How many ordered triplets $(x,y,z)$ exist that are primitive inverse-cubed triplets?

If you enjoyed this problem, you might also want to check out this and this problem.