Find The Length

Let $\triangle ABC$ be a triangle with side lengths $BC = 9, CA= 8, AB= 6.$ Let $O$ and $I$ be the circumcenter and incenter of $\triangle ABC$ respectively. The incircle of $\triangle ABC$ touches $BC$ at point $D.$ Let $X$ be the second point of intersection of line $AO$ and the circumcircle of $\triangle AID$ (apart from $A$). Given that $AX = \sqrt{\dfrac{a}{b}}$ for some coprime positive integers $a,b,$ find $a+b.$

Details and assumptions

• The image shown is not accurate.

• This problem is adapted from a Russia 10th grade geometry problem.