
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
Determine the maximal size of the set  $S$  such that:
i) all elements of  $S$  are natural numbers not exceeding  $100$ ;
ii) for any two elements  $a,b$  in  $S$ , there exists  $c$  in  $S$  such that  $(a,c)=(b,c)=1$ ;
iii) for any two elements  $a,b$  in  $S$ , there exists  $d$  in  $S$  such that  $(a,d)>1,(b,d)>1$ .

*Yao Jiangang*

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