
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 two unequal non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, the vector groups $(\overrightarrow{x_{1}}, \overrightarrow{x_{2}}, \overrightarrow{x_{3}}, \overrightarrow{x_{4}})$ and $(\overrightarrow{y_{1}}, \overrightarrow{y_{2}}, \overrightarrow{y_{3}}, \overrightarrow{y_{4}})$ are both formed by arranging 2 $\overrightarrow{a}$ and 2 $\overrightarrow{b}$. Let $S= \overrightarrow{x_{1}}\cdot \overrightarrow{y_{1}}+ \overrightarrow{x_{2}}\cdot \overrightarrow{y_{2}}+ \overrightarrow{x_{3}}\cdot \overrightarrow{y_{3}}+ \overrightarrow{x_{4}}\cdot \overrightarrow{y_{4}}$, then the minimum possible value of $S$ is _______ (expressed in terms of vectors $\overrightarrow{a}$ and $\overrightarrow{b}$).

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