HMMT November 2024 U23

Consider a quarter-circle with center $O$, arc $\widehat{A B}$, and radius 2. Draw a semicircle with diameter $\overline{O A}$ lying inside the quarter-circle. Points $P$ and $Q$ lie on the semicircle and segment $\overline{O B}$, respectively, such that line $P Q$ is tangent to the semicircle. As $P$ and $Q$ vary, compute the maximum possible area of triangle $B Q P$.