
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 $\overrightarrow{b}$, and two sets of vectors $\overrightarrow{x_{1}}$, $\overrightarrow{x_{2}}$, $\overrightarrow{x_{3}}$, $\overrightarrow{x_{4}}$, $\overrightarrow{x_{5}}$ and $\overrightarrow{y_{1}}$, $\overrightarrow{y_{2}}$, $\overrightarrow{y_{3}}$, $\overrightarrow{y_{4}}$, $\overrightarrow{y_{5}}$, each composed of 2 $\overrightarrow{a}$'s and 3 $\overrightarrow{b}$'s. Let $S = \overrightarrow{x_{1}} \cdot \overrightarrow{y_{1}} + \overrightarrow{x_{2}} \cdot \overrightarrow{y_{2}} + \overrightarrow{x_{3}} \cdot \overrightarrow{y_{3}} + \overrightarrow{x_{4}} \cdot \overrightarrow{y_{4}} + \overrightarrow{x_{5}} \cdot \overrightarrow{y_{5}}$, and $S_{min}$ represents the smallest possible value of $S$. The correct proposition(s) is/are _______ (write out the number(s) of all correct propositions).

① $S$ has 5 distinct values;
② If $\overrightarrow{a} \perp \overrightarrow{b}$, then $S_{min}$ is independent of $|\overrightarrow{a}|$;
③ If $\overrightarrow{a} // \overrightarrow{b}$, then $S_{min}$ is independent of $|\overrightarrow{b}|$;
④ If $|\overrightarrow{b}| > 4|\overrightarrow{a}|$, then $S_{min} > 0$;
⑤ If $|\overrightarrow{b}| = 2|\overrightarrow{a}|$ and $S_{min} = 8|\overrightarrow{a}|^{2}$, then the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\frac{\pi}{4}$.

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