
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 \( S = \{1,2, \cdots, 50\} \). Find the smallest natural number \( k \) such that any subset of \( S \) with \( k \) elements contains two distinct numbers \( a \) and \( b \) satisfying \( (a + b) \mid ab \).

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