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 \(b\).

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