
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

Nikolai and Petya are dividing $2n+1$ nuts, where $n \geqslant 2$, with each wanting to get as many as possible. The division proceeds through three stages in each of the three proposed methods:

1st stage: Petya divides all the nuts into two parts, with each part containing at least two nuts.

2nd stage: Nikolai then divides each part again into two, with each part containing at least one nut.

(Stages 1 and 2 are common to all three methods.)

3rd stage: In the first method, Nikolai takes the largest and smallest parts; in the second method, Nikolai takes both middle parts; in the third method, Nikolai either takes the largest and smallest parts or both middle parts, but has to give Petya one nut for the right to choose.

Determine which method is the most advantageous for Nikolai and which is the least advantageous for him.

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