Triply Complex Triples

Algebra Level 3

Consider all ordered triples of complex numbers $$(a, b, c)$$ which satisfy the following equations:

$\begin{cases} ab + 3b = - 9 \\ bc + 3c = - 9 \\ ca + 3a = - 9. \\ \end{cases}$

What is the largest possible value of $$\lvert abc \rvert$$?

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