
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 set \( S = \{1, 2, \ldots, 2022\} \) is to be partitioned into \( n \) disjoint subsets \( S_1, S_2, \ldots, S_n \) such that for each \( i \in \{1, 2, \ldots, n\} \), exactly one of the following statements is true:
(a) For all \( x, y \in S_i \) with \( x \neq y, \operatorname{gcd}(x, y) > 1 \).
(b) For all \( x, y \in S_i \) with \( x \neq y, \operatorname{gcd}(x, y) = 1 \).
Find the smallest value of \( n \) for which this is possible.

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