
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 $A_{2n}=(a_{1},a_{2},\ldots,a_{2n})$ be an ordered array consisting of $2n$ real numbers, satisfying the following conditions:  
1. $a_{i} \in \{1,-1\}$, $i=1,2,\ldots,2n$;  
2. $a_{1}+a_{2}+\ldots+a_{2n}=0$;  
3. $a_{1}+a_{2}+\ldots+a_{i} \geqslant 0$, $i=1,2,\ldots,2n-1$.  

(I) When $n=3$, list all $A_{6}$ that satisfy the given conditions;  
(II) Let $n=2k-1$, where $k \in \mathbb{N}^{*}$, find the set of possible values for $a_{1}+a_{2}+\ldots+a_{n}$;  
(III) Given a positive integer $n$, find the number of $A_{2n}$.

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