
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 positive integers \( m > 1 \) and \( n \), where both \( m \) and \( n \) are odd, and \( m \mid n \), let \( X \) be a non-empty subset of the set \(\{1, 2, \cdots, n\}\), and let \( S(X) \) be the sum of all elements in the set \( X \). Find the number of subsets \( X \) such that \( m \mid S(X) \).

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