Let f(z)=\frac{a z+b}{c z+d} for a, b, c, d \in \mathbb{C}. Suppose that f(1)=i, f(2)=i^{2}, and f(3)=i^{3}. If the real part of f(4) can be written as \frac{m}{n} for relatively prime positive integers m, n, find m^{2}+n^{2}.
Let f(z)=\frac{a z+b}{c z+d} for a, b, c, d \in \mathbb{C}. Suppose that f(1)=i, f(2)=i^{2}, and f(3)=i^{3}. If the real part of f(4) can be written as \frac{m}{n} for relatively prime positive integers m, n, find m^{2}+n^{2}.