Nemotron-IMO-Bench 2026 G41

Let $n>1$ and let $P$ be a point inside the cyclic $n$-gon $A_1\ldots A_{2n}$ such that the angles $PA_1A_2, PA_2A_3,\ldots, PA_{2n}A_1$ are equal. Prove that $$\prod_{i=1}^n A_{2i-1}A_{2i}=\prod_{i=1}^n A_{2i}A_{2i+1}.$$