I Love 315 to \(\infty\)

\[x^2\big(y^3+z^3\big)=315(xyz+7)\]

Let \((x_1, y_1, z_1), (x_2, y_2, z_2), \ldots , (x_n, y_n, z_n)\) be all of the solutions to the equation above, where \(x, y, z\) are positive integers. Find \(\displaystyle \sum_{k=1}^{n} (x_k+y_k+z_k) \).

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