
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 \( S = \left\{ m \mid m \in \mathbf{N}_{+}, m \right. \text{'s prime factors are all less than or equal to 10} \}. \) Find the smallest positive integer \( n \) such that in any subset of size \( n \) of \( S \), there exist 4 distinct numbers whose product is a perfect square.

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