
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  $A_0B_0+B_0C_0+C_0A_0=1$ . Then he chooses points  $A_1$ ,  $B_1$ ,  $C_1$  (not necessarily distinct) in such a way that  $A_1B_1=A_0B_0$  and  $B_1C_1=B_0C_0$ .
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  $A_3B_3=A_2B_2$  and  $B_3C_3=B_2C_2$ . What are the smallest and the greatest possible values of  $A_3B_3+B_3C_3+C_3A_3$  Bob can obtain?

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