
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

In the picture, several circles are drawn and connected by line segments. Kostya chooses a natural number \( n \) and places different natural numbers in the circles, not exceeding \( n \), so that the following property holds for all the numbers he places: if the numbers \( a \) and \( b \) are connected by a line 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, for the picture in Fig. 2, Kostya chose \( n=45 \) and found a suitable arrangement - it is shown in Fig. 3.

a) What is the smallest \( n \) such that the required number arrangement exists for Fig. 2?

b) Is it possible to arrange the numbers in the circles in Fig. 4 with \( n=25 \)?

c) Is it possible to arrange the numbers in the circles in Fig. 4 with \( n=39 \)?

d) What is the smallest \( n \) such that a number arrangement exists in the circles in Fig. 4?

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