
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 $\overset{→}{a}$ and $\overset{→}{b}$, the vector groups $(\overset{→}{{x}_{1}}, \overset{→}{{x}_{2}}, \overset{→}{{x}_{3}}, \overset{→}{{x}_{4}})$ and $(\overset{→}{{y}_{1}}, \overset{→}{{y}_{2}}, \overset{→}{{y}_{3}}, \overset{→}{{y}_{4}})$ are both formed by arranging 2 $\overset{→}{a}$ and 2 $\overset{→}{b}$. Let $S= \overset{→}{{x}_{1}}\cdot \overset{→}{{y}_{1}}+ \overset{→}{{x}_{2}}\cdot \overset{→}{{y}_{2}}+ \overset{→}{{x}_{3}}\cdot \overset{→}{{y}_{3}}+ \overset{→}{{x}_{4}}\cdot \overset{→}{{y}_{4}}$, then the minimum possible value of $S$ among all its possible values is _________. (Expressed in terms of vectors $\overset{→}{a}$ and $\overset{→}{b}$)

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