Complex Numbers and Quadratic Equations: JEE Main 2021 Mathematics question

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