
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
Let \( a_{1}, a_{2}, \ldots, a_{n} \) be positive integers, not necessarily distinct but with at least five distinct values. Suppose that for any \( 1 \leq i < j \leq n \), there exist \( k, \ell \), both different from \( i \) and \( j \) such that \( a_{i} + a_{j} = a_{k} + a_{\ell} \). What is the smallest possible value of \( n \)?

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