
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 an integer $n \geq 3$ and $n$ numbers $a_{1}, a_{2}, \cdots, a_{n}$, define the pairwise sums $b_{ij} = a_{i} + a_{j}$ for $i > j$. Determine the minimum value of $k$ such that selecting any $k$ numbers from this table uniquely determines the $n$ numbers $a_{1}, a_{2}, \cdots, a_{n}$.

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