Acute triangle $A_1A_2A_3$ is inscribed in a circle of radius $2$. Prove that one can choose points $B_1, B_2, B_3$ on the arcs $A_1A_2, A_2A_3, A_3A_1$ respectively, such that the numerical value of the area of the hexagon $A_1B_1A_2B_2A_3B_3$ is equal to the numerical value of the perimeter of the triangle $A_1A_2A_3.$