
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 \( N \) be a natural number greater than 3 that has the following property: no matter how the set \( S_n = \{1, 2, \cdots, n\} \) is partitioned into two groups, there is always a group that contains three numbers \( a, b, c \) (where \( a \) and \( b \) are allowed to be equal) such that \( ab = c \). Find the smallest such \( n \).

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