Let z and w be two complex numbers such that w=z z̄-2 z+2,|(z+i)/(z-3 i)|=1 and Re (w) has minimum value. Then, the minimum value of n ∈ N for which…

Let $\displaystyle z$ and $\displaystyle w$ be two complex numbers such that $\displaystyle w=z \bar{z}-2 z+2,\left|\frac{z+i}{z-3 i}\right|=1$ and $\displaystyle \operatorname{Re}(w)$ has minimum value. Then, the minimum value of $\displaystyle \mathrm{n} \in \mathrm{N}$ for which $\displaystyle w^{\mathrm{n}}$ is real, is equal to \_\_\_\_.