AIME 2016b Problem 3

Let $x,y,$ and $z$ be real numbers satisfying the system \begin{align*} \log_2(xyz-3+\log_5 x)&=5,\ \log_3(xyz-3+\log_5 y)&=4,\ \log_4(xyz-3+\log_5 z)&=4.\ \end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.