CMO 2022 Problem 1

Let $a$ and $b$ be two positive real numbers, and $AB$ a segment of length $a$ on a plane. Let $C,D$ be two variable points on the plane such that $ABCD$ is a non-degenerate convex quadrilateral with $BC=CD=b$ and $DA=a$ . It is easy to see that there is a circle tangent to all four sides of the quadrilateral $ABCD$ . Find the precise locus of the point $I$ .