
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
Given any natural number \( n \) (where \( n > 3 \)), if the set \( S_{n} = \{3, 4, \cdots, n\} \) is divided into two groups arbitrarily, there will always be three numbers \( a, b, c \) (allowing \( a = b \)) in one of the groups such that \( ab = c \). Find the smallest value of \( n \) for which this property holds.

(Source: 1988 National Training Team Test, China)

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