
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 a sequence $A\_0$: $a\_0$, $a\_1$, $a\_2$, $…$, $a\_n(n∈N^{})$, apply transformation $T$ to obtain sequence $A\_1$: $a\_1+a\_2$, $a\_2+a\_3$, $…$, $a\_{n-1}+a\_n$, denoted as $A\_1=T(A\_0)$. Continue applying transformation $T$ to $A\_1$ to obtain sequence $A\_2=T(A\_1)=T(T(A\_0))$, denoted as $A\_2=T\_2(A\_0)$; $…$; $A\__{n-1}=T\__{n-1}(A\_0)$. The final sequence $A\__{n-1}$ has only one number, denoted as $S(A\_0)$.

1. If sequence $A\_0$ is 1, 2, 3, find $S(A\_0)$;
2. If sequence $A\_0$ is 1, 2, $…$, $n$, find $S(A\_0)$;
3. If sequences $A$ and $B$ are exactly the same, they are considered equal, denoted as $A=B$. If sequence $B$ is a permutation of sequence $A\_0$: 1, 2, $…$, $n$, what is the condition for $S(B)=S(A\_0)$? Please explain the reason.

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