Real single solution of an algebraic nonlinear system

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Inspired from "How.." I extract the following problem that I have solved:

Prove that $(x, y) = (1, -2)$ is the only real solution of the system $$\begin{cases}x^5+5(xy+1)(y-x^2)=16\\y^5+5(xy+1)(y^2+x)=-57\end{cases}$$