
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 natural numbers $n$ greater than 1 for which there exists a permutation $a_{1}, a_{2}, \ldots, a_{n}$ of the numbers $1, 2, 3, \ldots, n$ such that the products $a_{1}, a_{1} a_{2}, a_{1} a_{2} a_{3}, \ldots, a_{1} a_{2} \cdots a_{n}$ all give different remainders when divided by $n$.

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