OMCC 2022 Problem 4

Let $A_1A_2A_3A_4$ be a rectangle and let $S_1,S_2,S_3,S_4$ four circumferences inside of the rectangle such that $S_k$ and $S_{k+1}$ are tangent to each other and tangent to the side $A_kA_{k+1}$ for $k=1,2,3,4$ , where $A_5=A_1$ and $S_5=S_1$ . Prove that $A_1A_2A_3A_4$ is a square.