Question 13
Point $P$ lies on the diagonal $AC$ of square $ABCD$ with $AP \gt CP.$ Let $O_1$ and $O_2$ be the circumcenters of triangles $ABP$ and $CDP,$ respectively. Given that $AB = 12$ and $\angle O_1PO_2 = 120^\circ,$ then $AP = \sqrt{a} + \sqrt{b},$ where $a$ and $b$ are positive integers. Find $a + b.$