HMMT 2020 G7

Let $\Gamma$ be a circle, and $\omega_{1}$ and $\omega_{2}$ be two non-intersecting circles inside $\Gamma$ that are internally tangent to $\Gamma$ at $X_{1}$ and $X_{2}$, respectively. Let one of the common internal tangents of $\omega_{1}$ and $\omega_{2}$ touch $\omega_{1}$ and $\omega_{2}$ at $T_{1}$ and $T_{2}$, respectively, while intersecting $\Gamma$ at two points $A$ and $B$. Given that $2 X_{1} T_{1}=X_{2} T_{2}$ and that $\omega_{1}, \omega_{2}$, and $\Gamma$ have radii 2, 3, and 12 , respectively, compute the length of $A B$.