
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
Let $\overrightarrow{a}$ and $\overrightarrow{b}$ be non-zero vectors, where $|\overrightarrow{b}| = 2|\overrightarrow{a}|$. Four pairs of vectors $\overrightarrow{x_{1}}$, $\overrightarrow{x_{2}}$, $\overrightarrow{x_{3}}$, $\overrightarrow{x_{4}}$ and $\overrightarrow{y_{1}}$, $\overrightarrow{y_{2}}$, $\overrightarrow{y_{3}}$, $\overrightarrow{y_{4}}$ are composed of 2 $\overrightarrow{a}$'s and 2 $\overrightarrow{b}$'s. If the minimum possible value of $\overrightarrow{x_{1}} \cdot \overrightarrow{y_{1}} + \overrightarrow{x_{2}} \cdot \overrightarrow{y_{2}} + \overrightarrow{x_{3}} \cdot \overrightarrow{y_{3}} + \overrightarrow{x_{4}} \cdot \overrightarrow{y_{4}}$ is $4|\overrightarrow{a}|^{2}$, determine the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$.

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