
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

On the image, there are several circles connected by segments. Kostya chooses a natural number \( n \) and places different natural numbers not exceeding \( n \) in the circles so that for all the placed numbers the following property is satisfied: if the numbers \( a \) and \( b \) are connected by a segment, then the difference \( a - b \) must be coprime with \( n \); if they are not connected, then the numbers \( a - b \) and \( n \) must have a common natural divisor greater than 1. For example, in the image (fig. 2) Kostya took \( n = 75 \) and arranged the numbers accordingly, as shown in fig. 3.

a) What is the smallest \( n \) for which the required arrangement of numbers exists in fig. 2?

b) Is it possible to place the numbers in the circles in fig. 4 for \( n = 49 \)?

c) Is it possible to place the numbers in the circles in fig. 4 for \( n = 33 \)?

d) What is the smallest \( n \) for which the arrangement of numbers exists in the circles in fig. 4?

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