
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  $S$  be a set of positive integers, each of them having exactly  $100$  digits in base  $10$  representation. An element of  $S$  is called *atom* if it is not divisible by the sum of any two (not necessarily distinct) elements of  $S$ . If  $S$  contains at most  $10$  atoms, at most how many elements can  $S$  have?

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