
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
\( f \) is a mapping from the set of natural numbers \( N \) to the set \( A \). If for \( x, y \in \mathbf{N} \), \( x - y \) is a prime number, it always holds that \( f(x) \neq f(y) \). How many elements must \( A \) have at least?

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