
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
S is a subset of {1, 2, 3, ..., 100} such that given any two elements \( a, b \) in S, there is an element \( c \) in S coprime to \( a \) and \( b \), and there is an element \( d \) in S which has a common factor with \( a \) and a common factor with \( b \). What is the largest possible number of elements in S?

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