Primitive Harmonic Triplets

Define a "primitive harmonic triplet" as any ordered triplet $(x,y,z),$ where $x,y,z\in\mathbb{Z}^+$ and $\gcd{(x,y,z)}=1,$ such that $\frac{1}{x}+\frac{1}{y}=\frac{1}{z}.$ Are there infinitely many primitive harmonic triplets?

• If the answer is no, enter the number of ordered pairs $(x,y)$ such that $(x,y,6)$ is a primitive harmonic triplet.
• If the answer is yes, enter the number of ordered pairs $(x,y)$ such that $\big(x,\ y,\ 2^{5}\times 3^{4}\times 5^{3}\times 7^{2}\times 11\big)$ is a primitive harmonic triplet.

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