
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
Find all positive integers \( n \) such that there exists a real-coefficient polynomial \( f(x) \) satisfying the following two properties:
1. For any integer \( k \), \( f(k) \) is an integer if and only if \( k \) is not divisible by \( n \);
2. The degree of the polynomial \( f(x) \) is less than \( n \).

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