
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 \( n > 2 \), construct a set of \( n \) distinct numbers \( a_{1}, a_{2}, \ldots, a_{n} \) such that the set
\[
S = \{a_{i} + a_{j} \mid 1 \leq i < j \leq n \}
\]
contains the minimum possible number of distinct sums.

Also, construct a set of \( n \) distinct numbers \( b_{1}, b_{2}, \ldots, b_{n} \) such that the set
\[
T = \{ b_{i} + b_{j} \mid 1 \leq i < j \leq n \}
\]
contains the maximum possible number of distinct sums.

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