Primitive Inverse-Cubed Triplets!

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

1x3+1y3=1z3.\frac{1}{x^3}+\frac{1}{y^3}=\frac{1}{z^3}.

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

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