
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 positive integers  $n \geqslant 2$  for which there exist  $n$  real numbers  $a_1<\cdots<a_n$  and a real number  $r>0$  such that the  $\tfrac{1}{2}n(n-1)$  differences  $a_j-a_i$  for  $1 \leqslant i<j \leqslant n$  are equal, in some order, to the numbers  $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$ .

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