Let $w$ be an arbitrary complex number with a non-zero imaginary part and a non-zero real part. There are $n$ values of $k \in \{0, 1, 2, 3, 4, . . . , 999\}$ for which $$z =\left( \overline{w}w^3 - w \overline{w}^3\right)^k$$is a real number. Enter $n$.