
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 positive integer $n \geq 2$, and a set $S$ containing $n$ pairwise distinct real numbers $a_{1}, a_{2}, \ldots, a_{n}$, define $k(S)$ as the number of distinct numbers of the form $a_{i} + 2^{j}$ for $i, j = 1, 2, \ldots, n$. Find the minimum possible value of $k(S)$ for all possible sets $S$.

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