To bound or not to bound!

Number Theory Level 5

Consider the following diophantine equation in natural numbers\[x+y^2+z^3=xyz\]where \(z=\gcd(x,y)\). Let \((x_1,y_1,z_1), (x_2,y_2,z_2), ..., (x_n,y_n,z_n)\) be the all triples which satisfy the equation. Find\[\sum_{k=1}^{n} (x_k+y_k+z_k).\]

Details and Assumptions

\(\gcd(a,b)\) is the greatest common divisor of \(a\) and \(b\).

This is from IMO1995 shortlist and belongs to this set.

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