
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
During a school recess, \( n \) children are sitting in a circle, playing a game. A teacher walks clockwise around the circle, distributing candies to the children according to the following rule:

1. Pick a child and give them and the next child (clockwise) one candy each.
2. Skip the next child, and give the following child one candy.
3. Skip the next two children, and give the following child one candy.
4. Continue this pattern (skip three children, give one candy, and so on).

Determine the value of \( n \) such that all children eventually (possibly after the teacher makes many rounds) receive at least one candy.

(Note: This problem was part of the 31st International Mathematical Olympiad shortlist and the 1990 Asia-Pacific Mathematical Olympiad.)

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