
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 \( n \in \mathbb{N} \) be greater than 3, and it has the following property: no matter how the set \( S_{n} = \{1, 2, \cdots, n\} \) is divided into two groups, there is always some group that contains three numbers \( a, b, c \) (allowing \( a = b \)) such that \( ab = c \). Find the smallest value of such \( n \).

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