Let $a, b, c, x, y,$ and $z$ be complex numbers such that:
\[ a = \frac{b+c}{x-2}, \quad b = \frac{c+a}{y-2}, \quad c = \frac{a+b}{z-2}. \]
Given that:
\[ xy + yz + zx = 1000 \quad \text{and} \quad x + y + z = 2016, \]
find the value of $xyz$.

Let's think step by step and output the final answer within \boxed{}.
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