
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
Determine the smallest possible value of the expression   $$ \frac{ab+1}{a+b}+\frac{bc+1}{b+c}+\frac{ca+1}{c+a} $$   where  $a,b,c \in \mathbb{R}$  satisfy  $a+b+c = -1$  and  $abc \leqslant -3$ 

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