
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 each integer  $n > 1$ , find a set of  $n$  integers  $\{a_1, a_2,..., a_n\}$  such that the set of numbers  $\{a_1+a_j | 1 \le  i \le j \le n\}$  leave distinct remainders when divided by  $n(n + 1)/2$ . If such a set of integers does not exist, give a proof.

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