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