Let x, y, and z be real numbers satisfying the system
\begin{aligned}
& \log _{2}\left(x y z-3+\log _{5} x\right)=5 \\
& \log _{3}\left(x y z-3+\log _{5} y\right)=4 \\
& \log _{4}\left(x y z-3+\log _{5} z\right)=4
\end{aligned}
Find the value of \left|\log _{5} x\right|+\left|\log _{5} y\right|+\left|\log _{5} z\right|.