Find The Length

Let $A$ and $B$ be two distinct points on a circle $\Gamma$ that are not diametrically opposite. Let $B'$ be the diametrically opposite point of $B$ on $\Gamma.$ Let $C$ be the point on $\Gamma$ apart from $B$ such that $B'A= AC,$ and let $C'$ be the diametrically opposite point of $C$ on $\Gamma.$ If $AC'= 5,$ find the distance between $A$ and $B.$

Details and assumptions

• The diametrically opposite point of $X$ on a circle $\omega$ is the unique point $X'$ on $\omega$ apart from $X$ such that $XX'$ is a diameter of $\omega.$