It's "How Many", But Not An Inequality

Algebra Level 3

{x3+y3=1x4+y4=1\large\begin{cases} \sqrt[3]{x}+\sqrt[3]{y}=1\\ \sqrt[4]{x}+\sqrt[4]{y}=1\end{cases}

How many real pairs of (x,y)(x,y) satisfy the system above?

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