
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
A set \( S \) of \( k \) distinct integers \( n_i \) is such that \( \prod n_i \) divides \( \prod (n_i + m) \) for all integers \( m \). Must 1 or -1 belong to \( S \)? If all members of \( S \) are positive, is \( S \) necessarily just \(\{1, 2, \ldots, k\}\)?

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