GAIC Math 2024 Problem 93

Given that the vertices of triangle $\triangle OAB$ are $O(0,0)$, $A(4,4\sqrt{3})$, and $B(8,0)$, with the incenter denoted as $I$, let $\Gamma$ be a circle passing through points $A$ and $B$, intersecting circle $\odot I$ at points $P$ and $Q$. If the tangents drawn through points $P$ and $Q$ are perpendicular, then the radius of circle $\Gamma$ is _____.