3 variables, 2 equations, 1 answer

\[\large x + y = z^2\\ \large x^2 + y^2 = z^3\]

If positive integral solutions \((x_1, y_1, z_1), (x_2, y_2, z_2), \ldots, (x_n, y_n, z_n)\) satisfy the system of equations above, find the value of

\[\large \displaystyle \sum_{i=1}^n (x_i + y_i + z_i)\]

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