RMO 2015!

Algebra Level 4

$\begin{cases} \begin{aligned} (1 + x)\big(1 + x^2\big)\big(1 + x^4\big) &= 1 + y^7 \\\\ (1 + y)\big(1 + y^2\big)\big(1 + y^4\big) &= 1 + x^7 \end{aligned} \end{cases}$

Find the number of ordered pairs of real numbers $(x, y)$ for which the above system of equations is satisfied.