# 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?