$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$

For non-negative integer $n$, write
$$(1+4\sqrt[3]2-4\sqrt[3]4)^n=a_n+b_n\sqrt[3]2+c_n\sqrt[3]4$$
where $a_n,b_n,c_n$ are integers. For any non-negative integer $m$, prove
or disprove
$$2^{m+2}|c_n\iff2^m|n$$
So far I have $[a_n, b_n, c_n]^T=[1, -8, 8; 4, 1, -8; -4, 4, 1][a_{n-1},
b_{n-1}, c_{n-1}]^T$ where [1, -8, 8; 4, 1, -8; -4, 4, 1] is a 3 by 3
matrix with rows $(1, -8, 8) ; (4, 1, -8); (-4, 4, 1)$