ÖMO 2006 Problem 6

Let $A$ be an integer not equal to $0$ . Solve the following system of equations in $\mathbb{Z}^3$ . $x + y^2 + z^3 = A$ $\frac {1}{x} + \frac {1}{y^2} + \frac {1}{z^3} = \frac {1}{A}$ $xy^2z^3 = A^2$