
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  $S=\{1,2,\ldots ,98\}$ . Find the least natural number  $n$  such that we can pick out  $10$  numbers in any  $n$ -element subset of  $S$  satisfying the following condition: no matter how we equally divide the  $10$  numbers into two groups, there exists a number in one group such that it is coprime to the other numbers in that group, meanwhile there also exists a number in the other group such that it is not coprime to any of the other numbers in the same group.

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