Nickolas and Peter divide $2n+1$ nuts amongst each other. Both of them want to get as many as possible. Three methods are suggested to them for doing so, each consisting of three stages. The first two stages are the same in all three methods: [i]Stage 1:[/i] Peter divides the nuts into 2 heaps, each containing at least 2 nuts. [i]Stage 2:[/i] Nickolas divides both heaps into 2 heaps, each containing at least 1 nut. Finally, stage 3 varies among the three methods as follows: [i]Method 1:[/i] Nickolas takes the smallest and largest of the heaps. [i]Method 2:[/i] Nickolas takes the two middle size heaps. [i]Method 3:[/i] Nickolas chooses between taking the biggest and the smallest heap or the two middle size heaps, but gives one nut to Peter for the right of choice. Determine the most and the least profitable method for Nickolas.