
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
a) Does there exist an infinite sequence of natural numbers with the following property: none of these numbers divide each other, but among any three numbers, we can choose two such that their sum is divisible by the third?

b) If not, how many numbers can there be in a set with such a property?

c) Solve the same problem with the additional condition: the set is allowed to include only odd numbers. Here is an example of such a set of four numbers: $3, 5, 7, 107$. In the triplet $3, 5, 7$ the sum $5 + 7$ is divisible by 3; in the triplet $5, 7, 107$ the sum $107 + 5$ is divisible by 7; in the triplet $3, 7, 107$ the sum $7 + 107$ is divisible by 3; finally, in the triplet $3, 5, 107$ the sum $3 + 107$ is divisible by 5.

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