
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 distinct 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 arranged from $2$ $\overrightarrow{a}$'s and $2$ $\overrightarrow{b}$'s. Let $S=\overrightarrow{x_1} \cdot \overrightarrow{y_1}+\overrightarrow{x_2} \cdot \overrightarrow{y_2}+\overrightarrow{x_3} \cdot \overrightarrow{y_3}+\overrightarrow{x_4} \cdot \overrightarrow{y_4}$. Find the smallest possible value of $S$. (Express the answer using vectors $\overrightarrow{a}$ and $\overrightarrow{b}$.)

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