
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  $\mathbb{N}$  denote the set of positive integers, and let  $S$  be a set. There exists a function  $f :\mathbb{N} \rightarrow S$  such that if  $x$  and  $y$  are a pair of positive integers with their difference being a prime number, then  $f(x) \neq f(y)$ . Determine the minimum number of elements in  $S$ .

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