RMO 2015!

Algebra Level 4

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

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