In the coordinate plane, right triangle $ABC$ has vertices $A(0,0)$, $B(x,y)$, and $C(x,0)$ for some positive $x$ and $y$. Another right triangle $DEF$ has vertices $D(0,0)$, $E(u,v)$, and $F(u,0)$ for some positive $u$ and $v$. If $△ABC$ is similar to $△DEF$, $\angle C$ and $\angle F$ are right angles, and $\tan (A)=\frac{12}{5}$, what is the value of $\tan (E)$?