
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 \( M = \{2, 3, \cdots, 1000\} \). Find the smallest natural number \( n \) such that in any subset of \( M \) with \( n \) elements, there exist three disjoint 4-element subsets \( S, T, U \) satisfying the following three conditions:
1. For any two elements in \( S \), the larger number is a multiple of the smaller number. The sets \( T \) and \( U \) have the same property.
2. For any \( s \in S \) and \( t \in T \), \((s, t) = 1\).
3. For any \( s \in S \) and \( u \in U \), \((s, u) > 1\).

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