
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 a positive integer \( n \), an allowed move is to form \( 2n + 1 \) or \( 3n + 2 \). The set \( S_n \) is the set of all numbers that can be obtained by a sequence of allowed moves starting with \( n \). For example, we can form \( 5 \rightarrow 11 \rightarrow 35 \) so 5, 11, and 35 belong to \( S_5 \). We call \( m \) and \( n \) compatible if \( S_m \cap S_n \) is non-empty. Which members of \(\{1, 2, 3, \ldots, 2002\}\) are compatible with 2003?

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