Triangle $ABC$ is an isosceles right triangle with $AB=AC=4$. Let $M$ be the midpoint of hypotenuse $\overline{BC}$. Points $I$ and $E$ lie on sides $\overline{AC}$ and $\overline{AB}$, respectively so that $AI>AE$ and $AIME$ is a cyclic quadrilateral. Given that triangle $EMI$ has area $3$, the length $CI$ can be written as $a-\sqrt{b}$, where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime. What is the value of $a+b$?