Let S ={z ∈ C:|z-2| ≤ 1, z(1+i)+z̄(1-i) ≤ 2}. Let |z-4 i| attains minimum and maximum values, respectively, at z_1 ∈ S and z_2 ∈ S. If…

Let $\displaystyle \mathrm{S}=\{z \in \mathbf{C}:|z-2| \leq 1, z(1+i)+\bar{z}(1-i) \leq 2\}$. Let $\displaystyle |z-4 i|$ attains minimum and maximum values, respectively, at $\displaystyle z_1 \in \mathrm{~S}$ and $\displaystyle z_2 \in \mathrm{~S}$. If $\displaystyle 5\left(\left|z_1\right|^2+\left|z_2\right|^2\right)=\alpha+\beta \sqrt{5}$, where $\displaystyle \alpha$ and $\displaystyle \beta$ are integers, then the value of $\displaystyle \alpha+\beta$ is equal to \_\_\_\_.