
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
The Fibonacci numbers \( F_{0}, F_{1}, F_{2}, \ldots \) are defined inductively by \( F_{0}=0, F_{1}=1 \), and \( F_{n+1} = F_{n} + F_{n-1} \) for \( n \geq 1 \). Given an integer \( n \geq 2 \), determine the smallest size of a set \( S \) of integers such that for every \( k = 2, 3, \ldots, n \) there exist some \( x, y \in S \) such that \( x - y = F_{k} \).

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