Let $S=\{z \in C:|z-2| \leq 1, z(1+i)+\bar{z}(1-i) \leq 2\}$. Let $|z-4 i|$ attains minimum and maximum values, respectively, at $\mathrm{z}_1 \in \mathrm{~S}$ and $\mathrm{z}_2 \in \mathrm{~S}$. If $5\left(\left|\mathrm{z}_1\right|^2+\left|\mathrm{z}_2\right|^2\right)=\alpha+\beta \sqrt{5}$, where $\alpha$ and $\beta$ are integers, then the value of $\alpha+\beta$ is equal to $\_\_\_\_$