The complex numbers z and w satisfy the system

\begin{aligned}
z+\frac{20 i}{w} & =5+i \\
w+\frac{12 i}{z} & =-4+10 i
\end{aligned}

Find the smallest possible value of |z w|^{2}.

The complex numbers z and w satisfy the system

\begin{aligned}
z+\frac{20 i}{w} & =5+i \\
w+\frac{12 i}{z} & =-4+10 i
\end{aligned}

Find the smallest possible value of |z w|^{2}.