Kashyap's system of equations

How many ordered triples of integer solutions \((x,y,z)\) are there to the following system of equations:

\[\begin{align} x + y & = 1 − z, \\ x^3 + y^3 & = 1 − z^2,\\ z & \ne 1 ? \end{align} \]

This problem is shared by Kashyap M.

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