HMMT 2006 S15

Let $a, b, c, d$ be real numbers so that $c, d$ are not both 0 . Define the function

$$m(x)=\frac{a x+b}{c x+d}$$

on all real numbers $x$ except possibly $-d / c$, in the event that $c \neq 0$. Suppose that the equation $x=m(m(x))$ has at least one solution that is not a solution of $x=m(x)$. Find all possible values of $a+d$. Prove that your answer is correct.