Complex Numbers and Quadratic Equations: JEE Main 2022 Mathematics question

Let $S = \{ z \in C:|z - 2| \le 1,\,z(1 + i) + \overline z (1 - i) \le 2\}$. Let $|z - 4i|$ attains minimum and maximum values, respectively, at z₁ $\in$ S and z₂ $\in$ S. If $5(|{z_1}{|^2} + |{z_2}{|^2}) = \alpha + \beta \sqrt 5$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha$ + $\beta$ is equal to ___________.