HMMT 2022 T3

Let triangle $A B C$ be an acute triangle with circumcircle $\Gamma$. Let $X$ and $Y$ be the midpoints of minor arcs $\widehat{A B}$ and $\widehat{A C}$ of $\Gamma$, respectively. If line $X Y$ is tangent to the incircle of triangle $A B C$ and the radius of $\Gamma$ is $R$, find, with proof, the value of $X Y$ in terms of $R$.