
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 $n$ be a positive integer and let $A=\{\alpha|\alpha=(t_1,t_2,\dots,t_n), t_k \in \{0,1\}, k=1,2,\dots,n\}$. For any elements $\alpha=(x_1,x_2,\dots,x_n)$ and $\beta=(y_1,y_2,\dots,y_n)$ in set $A$, define
$$
M(\alpha,\beta)=\frac{1}{2}\left[(x_1+y_1-|x_1-y_1|)+(x_2+y_2-|x_2-y_2|)+\dots+(x_n+y_n-|x_n-y_n|)\right]
$$
(I) When $n=3$ and $\alpha=(1,1,0)$, $\beta=(0,1,1)$, find the values of $M(\alpha,\alpha)$ and $M(\alpha,\beta)$.  
(II) When $n=4$, let $B$ be a subset of $A$ that satisfies: for any two elements $\alpha$, $\beta$ in $B$, $M(\alpha,\beta)$ is odd when $\alpha$ and $\beta$ are the same; $M(\alpha,\beta)$ is even when $\alpha$ and $\beta$ are different. Find the maximum number of elements in set $B$.  
(III) Given $n\geq 2$, let $B$ be a subset of $A$ such that for any two distinct elements $\alpha$, $\beta$ in $B$, $M(\alpha,\beta)=0$. Construct a subset $B$ with the maximum number of elements and explain the reasoning.

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