
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  $S = \{1, 2, \dots, 2022\}$  is to be partitioned into  $n$  disjoint subsets  $S_1, S_2, \dots, S_n$  such that for each  $i \in \{1, 2, \dots, n\}$ , exactly one of the following statements is true:

(a) For all  $x, y \in S_i$ , with  $x \neq y, \gcd(x, y) > 1.$ (b) For all  $x, y \in S_i$ , with  $x \neq y, \gcd(x, y) = 1.$ Find the smallest value of  $n$  for which this is possible.

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