Let \(f(z)= \frac{z+a}{z+b}\) and \(g(z)=f(f(z))\), where \(a\) and \(b\) are complex numbers. Suppose that \(\left| a \right| = 1\) and \(g(g(z))=z\) for all \(z\) for which \(g(g(z))\) is defined. Find the difference between the largest and smallest possible values of \(\left| b \right|\). The original answer is in the form \(\sqrt{m} - n\). Please give the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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