
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 such that \( n \geq 2 \). Given distinct real numbers \( a_{1}, a_{2}, \cdots, a_{n} \) forming a set \( S \), define \( k(S) \) as the number of distinct values in the form \( a_{i}+2^{j} \) for \( i, j = 1, 2, \ldots, n \). Determine the minimum possible value of \( k(S) \) for all sets \( S \).

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