INMO 2024 Problem 5

Let points $A_{1}, A_{2}$, and $A_{3}$ lie on the circle $\Gamma$ in counter-clockwise order, and let $P$ be a point in the same plane. For $i \in\{1,2,3\}$, let $\tau_{i}$ denote the counter-clockwise rotation of the plane centred at $A_{i}$, where the angle of the rotation is equal to the angle at vertex $A_{i}$ in $\triangle A_{1} A_{2} A_{3}$. Further, define $P_{i}$ to be the point $\tau_{i+2}\left(\tau_{i}\left(\tau_{i+1}(P)\right)\right)$, where indices are taken modulo 3 (i.e., $\tau_{4}=\tau_{1}$ and $\tau_{5}=\tau_{2}$ ).

Prove that the radius of the circumcircle of $\triangle P_{1} P_{2} P_{3}$ is at most the radius of $\Gamma$.