3 variables, 2 equations, 1 answer

Number Theory Level 4

$\large x + y = z^2\\ \large x^2 + y^2 = z^3$

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

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