
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 \( n (n \geqslant 2) \) such that there exist real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) making the set \( A=\left\{\mid a_{i}-a_{j}\right\} \mid 1 \leqslant i<j \leqslant n \} \) have exactly \( \frac{n(n-1)}{2} \) elements.

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