
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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?

Present the answer in LaTex format: \boxed{Your answer}
user