When does this even happen?

Let $ABC$ be a triangle with $AB = 7, AC = 9, BC = 10$ , circumcentre $O$ , circumradius $R$ , and circumcircle $\omega$ . Let the tangents to $\omega$ at $B, C$ meet at $X$ . A variable line $\ell$ passes through $O$ . Let $A_1$ be the projection of $X$ onto $\ell$ and $A_2$ be the reflection of $A_1$ over $O$ .

Suppose that there exist two points $Y, Z$ on $\ell$ such that $\angle YAB + \angle YBC + \angle YCA = \angle ZAB + \angle ZBC + \angle ZCA = 90^{\circ}$ , where all angles are directed, and furthermore that $O$ lies inside segment $YZ$ with $OY \cdot OZ = R^2$ . Then there are several possible values for the sine of the angle at which the angle bisector of $\angle AA_2O$ meets $BC$ .

If the product of these values can be expressed in the form $\frac{a\sqrt{b}}{c}$ for positive integers $a$ , $b$ , $c$ , with $b$ square-free and $a, c$ co-prime, determine $a+b+c$ .