Question 91
On a plane, there are given points $A_0, B_0, C_0$ (not necessarily distinct) such that $A_0B_0 + B_0C_0 + C_0A_0 = 1$. Points $A_1, B_1, C_1$ (not necessarily distinct) are chosen in such a way that $A_1B_1 = A_0B_0$ and $B_1C_1 = B_0C_0$. Points $A_2, B_2, C_2$ are chosen as a permutation of points $A_1, B_1, C_1$. Finally, points $A_3, B_3, C_3$ (not necessarily distinct) are chosen in such a way that $A_3B_3 = A_2B_2$ and $B_3C_3 = B_2C_2$. Find the least and the greatest possible value of $A_3B_3 + B_3C_3 + C_3A_3$.