
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
Find all values \( n \in \mathbf{N} \) for which there exists a permutation

\[ 
\left(a_{1} ; a_{2} ; \ldots ; a_{n}\right) 
\]

of the numbers \( 0,1, \ldots, n-1 \) such that all the numbers

\[ 
a_{1}, a_{1} a_{2}, a_{1} a_{2} a_{3}, \ldots, a_{1} a_{2} \ldots a_{n} 
\]

give different remainders when divided by \( n \).

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