A University of Cambridge Math interview question

Algebra Level 5

\[\begin{cases} x_1+x_2x_3x_4=2 \\ x_2+x_1x_3x_4=2 \\ x_3+x_1x_2x_4=2 \\ x_4+x_1x_2x_3=2 \end{cases} \]

Find the number of ordered quadruples of real numbers \((x_1,x_2,x_3,x_4) \) that satisfy the system of equations above.

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