
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
Given three points $A$, $B$, $C$ on a plane that satisfy $| \overrightarrow{AB}|= \sqrt {3}$, $| \overrightarrow{BC}|= \sqrt {5}$, $| \overrightarrow{CA}|=2 \sqrt {2}$, determine the value of $\overrightarrow{AB}\cdot \overrightarrow{BC}+ \overrightarrow{BC}\cdot \overrightarrow{CA}+ \overrightarrow{CA}\cdot \overrightarrow{AB}$.

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