
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 \( n \) for which there exists a permutation \( (a_1, a_2, \cdots, a_n) \) of \( 0, 1, 2, \cdots, n-1 \) such that the remainders when \( a_1, a_1 a_2, a_1 a_2 a_3, \cdots, a_1 a_2 \cdots a_n \) are divided by \( n \) are all different.

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