
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 the set $U_{n}=\{(x_{1},x_{2},\cdots,x_{n})|x_{i}\in \{0,1\}$, $i=1,2,\cdots,n, n\geqslant 3\}$, for any $\alpha =(x_{1},x_{2},\cdots,x_{n})\in U_{n}$, $\beta =(y_{1},y_{2},\cdots,y_{n})\in U_{n}$, define $\alpha *\beta =\max \{x_{1},y_{1}\}+\max \{x_{2},y_{2}\}+\cdots +\max \{x_{n},y_{n}\}$, where $\max \{a,b\}$ represents the maximum value between $a$ and $b$, for example, $\max \{1,0\}=1$, $\max \{1,1\}=1$.<br/>$(1)$ When $n=3$ and $\alpha =(0,1,0)$, list all elements $\beta$ that satisfy $\alpha *\beta =3$;<br/>$(2)$ Let $\alpha, \beta \in U_{n}$ satisfy $\alpha *\alpha +\beta *\beta =n$, find the maximum and minimum values of $\alpha *\beta$;<br/>$(3)$ If the subset $S$ of $U_{n}$ satisfies: for any $\{\alpha, \beta\}\subseteq S$, $\alpha *\beta \geqslant n$, find the maximum number $m_{S}$ of elements in set $S$.

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