Be realistic

Algebra Level 3

Let ff be a function defined for all complex numbers zz as f(z)=z22z+2f(z)=z^2-2z+2. Also let Img(z)\text{Img}(z) denote the imaginary part of zz and Re(z)\text{Re}(z) denote the real part of zz.

If a complex number zz is randomly selected such that Re(z){1,2,3,4}\text{Re}(z)\in\{1,2,3,4\} and Img(z){1,2,3,4,5,6}\text{Img}(z)\in\{1,2,3,4,5,6\}, then the probability that f(z)f(z) is real is ab\frac{a}{b}, where aa and bb are coprime, positive integers. Find a+ba+b.

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