
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
Let  $m,n$  be positive integers. Find the minimum positive integer  $N$  which satisfies the following condition. If there exists a set  $S$  of integers that contains a complete residue system module  $m$  such that  $| S | = N$ , then there exists a nonempty set  $A \subseteq S$  so that  $n\mid {\sum\limits_{x \in A} x }$ .

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