
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
Given the set \( S = \{1, 2, 3, \ldots, 2022\} \) and the set \( A = \{a_{1}, a_{2}, a_{3}, \ldots, a_{2022}\} \), which is a permutation of the set \( S \). It is known that for any \(1 \leq n, m \leq 2022\), the expression \( a_{n} + a_{m} \) is divisible by the greatest common divisor (GCD) of \( n \) and \( m \). Find the number of possible sets \( A \).

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