Nemotron-IMO-Bench 2026 G47

Let $A_1\ldots A_n$ be a convex $n$-gon inscribed in a circle with radius $1$, such that the center of the circle lies inside the $n$-gon. What is the smallest possible value of $\max (PA_1,\ldots, PA_n)$, when $P$ is a variable point in the plane of the $n$-gon?