2024 AIME II Problem 10

Let \triangle ABC have incenter I, circumcenter O, inradius 6, and circumradius 13. Suppose that \overline{IA} \perp \overline{OI}. Find AB \cdot AC.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.