
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
For which integers \( m > 1 \) does there exist a permutation \( a_{1}, a_{2}, \ldots, a_{m} \) of the numbers \( 1, 2, \ldots, m \) such that the sums \( a_{1}, a_{1} + a_{2}, \ldots, a_{1} + a_{2} + \ldots + a_{m} \) all give different remainders when divided by \( m \)?

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