Solutions and permutations

Algebra Level 5

\[ \left\{ \begin{array}{lcc} x^2+y^2+z^2 & = & 50 \\ (x+y)(y+z)(z+x) & = & 14 \\ x^3+y^3+z^3-3xyz& = & 146 \end{array} \right. \]

How many rational solutions does the system of equations above have?

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