
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 $n \geqslant 3$ be an integer. In a school, $n$ children are seated in a circle. Each child initially has a certain number of books. Whenever a child has at least two books and has strictly more books than the sum of the number of books of their two neighbors, they can make a redistribution by giving one book to each of their two neighbors. Every minute, if at least one child can make a redistribution, one of the children carries out a redistribution.

(i) For which integers $n \geqslant 3$ does there exist an initial distribution of books such that the children can continue redistributing books indefinitely?

(ii) Assume a child can only make a redistribution if they have at least 3 books. Reconsider the previous question under this condition.

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