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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