ILL 1989 Problem 22

$\forall n > 0, n \in \mathbb{Z},$ there exists uniquely determined integers $a_n, b_n, c_n \in \mathbb{Z}$ such

$$\left(1 + 4 \cdot \sqrt[3]{2} - 4 \cdot \sqrt[3]{4} \right)^n = a_n + b_n \cdot \sqrt[3]{2} + c_n \cdot \sqrt[3]{4}.$$

Prove that $c_n = 0$ implies $n = 0.$