
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

Given a natural number \( n \geq 5 \). Find:

1. For an \( n \)-element set \(\left\{a_1, a_2, \cdots, a_n\right\}\), how many distinct sums \( a_i + a_j \) (where \( 1 < i < j \leq n \)) are there at a minimum?
2. Identify all \( n \)-element sets that achieve this minimum number of distinct sums.

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