
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
$n$ boxes initially contain $1,2, \ldots, n$ marbles respectively ($n \geq 1$). Charlotte first adds a marble to each box. Then she adds a marble to each box in which the number of marbles is divisible by 2, then a marble to each box in which the number of marbles is divisible by 3, and so on, until she adds a marble to each box in which the number of marbles is divisible by $n$. For which values of $n$ does the procedure end with exactly $n+1$ marbles in every box?

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