Nemotron-IMO-Bench 2026 G14

A circle of radius $r$ is inscribed in triangle $ABC$. Circles with radii $r_1,r_2, r_3<r$ are inscribed between the angles $A,B,C$ and the incircle of triangle $ABC$. Prove that $$r_1+r_2+r_3 \geq r.$$