RO-TST 2013 Problem 1

Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that $$\left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}.$$ Prove that $c_{n}\equiv 1\pmod{3}$ if and only if $n\equiv 2\pmod{3}.$