
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 \( n \) be a positive integer \( n \geq 3 \), and given \( n \) numbers \( a_{1}, a_{2}, \cdots, a_{n} \), denote the pairwise sums as \( b_{ij} = a_{i} + a_{j} (i > j) \), resulting in the following table:
$$
\begin{array}{l}
b_{21} \\
b_{31} \quad b_{32} \\
\text {........................... } \\
b_{n1} \quad b_{n2} \ldots \ldots \ldots \ldots \ldots \ldots \ldots b_{n, n-1} \\
\end{array}
$$

If we choose \( k \) numbers from the table, and it is possible to uniquely determine the \( n \) numbers \( a_{1}, a_{2}, \cdots, a_{n} \), what is the minimum value of \( k \)?

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