Another interesting ratio

In isosceles triangle $ABC$ , $AB=AC=1$ while $BC=\frac{1}{2}$ . Points $P$ and $Q$ lie on $AB$ and $AC$ respectively. Two circles are constructed with $A$ , $P$ and $Q$ lying on one of them while points $B$ , $P$ , $Q$ and $C$ lie on the other circle. Interestingly, both circles have the same radius. If $AQ=\frac{a}{b}$ where $a$ and $b$ are positive coprime integers, $a+b=\boxed{?}$