
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
Unit vectors $\bar{e}_{1}, \bar{e}_{2}, \bar{e}_{3}$ satisfy the condition $\bar{e}_{1}+\bar{e}_{2}+\bar{e}_{3}=\overline{0}$. Find $\bar{e}_{1} \bar{e}_{2}+\bar{e}_{2} \bar{e}_{3}+\bar{e}_{3} \bar{e}_{1}$.

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