Variance of the real part of a complex random variables

Question

Let $S = \{ x : x^3 - 1 = 0, x \in \mathbf{C}\}$, denote the 3rd roots of unity. Let $c \in \mathbf{R}^n$ be a vector, and let $a^Tb = \sum_{i=1}^n a_i b_i$, for $a$, a real vector, and $b$ a complex vector. Let $X$ be the random vector with $X_i \sim \mathrm{Unif}(S)$, that is each of the roots in $S$ are taken with probability $1/3$. What is the variance of the real part of $(c^TX)^3$?