How do you solve this?

Algebra Level 3

\[\begin{cases} (1+x)(1+x^2)(1+x^4) = 1+y^7 \\ (1+y)(1+y^2)(1+y^4) = 1+x^7 \end{cases} \]

How many ordered pairs of real numbers \((x, y)\) satisfy the above equations?

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