Nemotron-IMO-Bench 2026 G44

Let $\Delta_0$ be an equilateral triangle with incircle $\omega$. A point on $\omega$ is reflected in the sides of $\Delta_0$ to obtain a new triangle $\Delta_1$. The same point is then reflected over the sides of $\Delta_1$ to obtain another triangle $\Delta_2$. Prove that the circumcircle of $\Delta_2$ is tangent to $\omega$.