HMMT 2024 G6

In triangle $A B C$, a circle $\omega$ with center $O$ passes through $B$ and $C$ and intersects segments $\overline{A B}$ and $\overline{A C}$ again at $B^{\prime}$ and $C^{\prime}$, respectively. Suppose that the circles with diameters $B B^{\prime}$ and $C C^{\prime}$ are externally tangent to each other at $T$. If $A B=18, A C=36$, and $A T=12$, compute $A O$.