Min Max Modulus Complex Number

Let S={z∈C:|z−2|≤1,z(1+i)+z(1−i)≤2}. Let |z−4i| attains minimum and maximum values, respectively, at z1∈S and z2∈S. If 5(z1|2+z2|2)=α+β√5, where α and β are integers, then the value of α+β is equal to_____