Let $$S = \{z \in C : |z - 2| \leq 1, z(1+i) + \bar{z}(1-i) \leq 2\}$$. Let $$|z - 4i|$$ attains minimum and maximum values, respectively, at $$z_1 \in S$$ and $$z_2 \in S$$. 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 ______.