Matrix finds the cube

Algebra Level 4

Let A=(abcbcacab)A = \left( \begin{matrix} a & b & c \\ b & c & a \\ c & a & b \end{matrix} \right) be a matrix where a,b,ca, b, c are complex numbers. If abc=1abc = 1 and ATA=IA^T A = I, what is the value of a3+b3+c3a^3 + b^3 + c^3?

Clarification: ATA^T is the transpose of AA. II is the identity matrix of order 3×33 \times 3.

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