A Good Triple

Algebra Level 5

A triple \((x, y, z) \in \mathbb{C^3}\) is called good iff the following conditions are satisfied. \[\begin{cases} (x+y+z) \left( x^3 + y^3 + z^3 + xyz \right) & = x^2 \left( x^2- y^2 \right) + y^2 \left( y^2 - z^2 \right) + z^2 \left( z^2 - x^2 \right) + 2014\\ 2xyz \left( \sqrt{xy}+\sqrt{yz}+\sqrt{zx} \right) & = 1007 \end{cases} \] How many good triples consisting of positive reals are there?

This problem appeared in the Proofathon Algebra contest, and was posed by me.
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