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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