
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  $K$  and  $N > K$  be fixed positive integers. Let  $n$  be a positive integer and let  $a_1, a_2, ..., a_n$  be distinct integers. Suppose that whenever  $m_1, m_2, ..., m_n$  are integers, not all equal to  $0$ , such that  $\mid{m_i}\mid \le K$  for each  $i$ , then the sum  $$ \sum_{i = 1}^{n} m_ia_i $$  is not divisible by  $N$ . What is the largest possible value of  $n$ ?

*Proposed by Ilija Jovcevski, North Macedonia*

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