Challenge · Medium problem

Positive real numbers $x,y,z$ satisfy $\log_2 x+\log_4 y+\log_4 z=2$, $\log_4 x+\log_2 y+\log_4 z=4$, and $\log_4 x+\log_4 y+\log_2 z=5$. Find $x+y+z$.

Give the exact value.