
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 \( \frac{1}{2} n(n-1) \) 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