HMMT 2018 G10

Let $A B C$ be a triangle such that $A B=6, B C=5, A C=7$. Let the tangents to the circumcircle of $A B C$ at $B$ and $C$ meet at $X$. Let $Z$ be a point on the circumcircle of $A B C$. Let $Y$ be the foot of the perpendicular from $X$ to $C Z$. Let $K$ be the intersection of the circumcircle of $B C Y$ with line $A B$. Given that $Y$ is on the interior of segment $C Z$ and $Y Z=3 C Y$, compute $A K$.