Let $x$, $y$, and $z$ be positive real numbers that satisfy\[2 \log_x (2y) = 2 \log_{2x} (4z) = \log_{2x^4} (8yz) \neq 0.\]the value of $xy^5 z$ can be expressed in the form $\frac{1}{2^{p/q}}$, where $p$ and $q$ are relatively prime positive integers. find $p + q$.