
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
Suppose that  $n$  people  $A_1$ ,  $A_2$ ,  $\ldots$ ,  $A_n$ , ( $n \geq 3$ ) are seated in a circle and that  $A_i$  has  $a_i$  objects such that 
\[ a_1 + a_2 + \cdots + a_n = nN \] 
where  $N$  is a positive integer. In order that each person has the same number of objects, each person  $A_i$  is to give or to receive a certain number of objects to or from its two neighbours  $A_{i-1}$  and  $A_{i+1}$ . (Here  $A_{n+1}$  means  $A_1$  and  $A_n$  means  $A_0$ .)  How should this redistribution be performed so that the total number of objects transferred is minimum?

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