Question 14
Let $\triangle ABC$ be an acute triangle with circumcenter $O$ and centroid $G.$ Let $X$ be the intersection of the line tangent to the circumcircle of $\triangle ABC$ at $A$ and the line perpendicular to $GO$ at $G.$ Let $Y$ be the intersection of lines $XG$ and $BC.$ Given that the measures of $\angle ABC,$ $\angle BCA,$ and $\angle XOY$ are in the ratio $13 : 2 : 17,$ the degree measure of $\angle BAC$ can be written as $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m + n.$