Let Z and W be two complex numbers such that $W=Z\overline{Z}-2Z+2,\left|\frac{Z+i}{Z-3i}\right|=1$ and $\mathrm{Re}\left(W\right)$ has
minimum value. Then the minimum value of ‘n’ for which  $W^n$ is real, is equal to ______

 $$\begin{gathered} Z=x+iy\Rightarrow W=x^2+y^2-2x-2iy+2 \\ \begin{array}{l}\mathrm{Re}\left(W\right)=x^2-2x+y^2+2\\ ={(x-1)}^2+y^2+\mathrm{1..........}\left(1\right)\end{array} \\ |Z+i|=|Z-3i|\Rightarrow y=\mathrm{1...........}\left(2\right) \end{gathered}$$
 Sub (2) in (1)  $\mathrm{Re}\left(W\right)={(x-1)}^2+2$
$\mathrm{Re}{\left(W\right)}_{\min }=2$   when  $x=1$
    $\therefore x=1,y=1$
  $\therefore Z=1+i\Rightarrow Z^n={(1+i)}^n$