
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

Determine all positive integers \( k \) for which there exist a positive integer \( m \) and a set \( S \) of positive integers such that any integer \( n > m \) can be written as a sum of distinct elements of \( S \) in exactly \( k \) ways.

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