
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
\( n \) (\( n \geq 3 \)) boxes are arranged in a circle, and balls are placed in the boxes in a counterclockwise direction as follows: 1 ball in the first box, 2 balls in the second box, and so on up to \( n \) balls in the \( n \)-th box. Then, the following operation is performed: Choose three consecutive boxes and either place 1 ball in each of the three boxes or remove 1 ball from each of the three boxes, given that none of the three boxes are empty. Determine all positive integers \( k \) such that it is possible, through a finite number of operations, for each of the \( n \) boxes to contain exactly \( k \) balls.

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