BALTIC-WAY 2020 Problem 15

On a plane, Bob chooses 3 points $A_{0}, B_{0}, C_{0}$ (not necessarily distinct) such that $A_{0} B_{0}+B_{0} C_{0}+C_{0} A_{0}=1$. Then he chooses points $A_{1}, B_{1}, C_{1}$ (not necessarily distinct) in such a way that $A_{1} B_{1}=A_{0} B_{0}$ and $B_{1} C_{1}=B_{0} C_{0}$. Next he chooses points $A_{2}, B_{2}, C_{2}$ as a permutation of points $A_{1}, B_{1}, C_{1}$. Finally, Bob chooses points $A_{3}, B_{3}, C_{3}$ (not necessarily distinct) in such a way that $A_{3} B_{3}=A_{2} B_{2}$ and $B_{3} C_{3}=B_{2} C_{2}$. What are the smallest and the greatest possible values of $A_{3} B_{3}+B_{3} C_{3}+C_{3} A_{3}$ Bob can obtain?