
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
In an equilateral triangle $ABC$ with side length $\sqrt{2}$, let $\overrightarrow{AB}= \overrightarrow{c}$, $\overrightarrow{BC}= \overrightarrow{a}$, and $\overrightarrow{CA}= \overrightarrow{b}$. Calculate the value of $\overrightarrow{a}\cdot \overrightarrow{b}+ \overrightarrow{b}\cdot \overrightarrow{c}+ \overrightarrow{c}\cdot \overrightarrow{a}$.

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