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

Let's think step by step and output the final answer within \boxed{}.
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