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A string can represent a natural number consisting of exactly $N$ digits. The following operation can be performed on this string: you may cut the string between any two consecutive digits, swap the two resulting parts without reversing them, and then reattach them. The string is called beautiful if, after this operation, the string still represents the same number. For example, the number $5656$ is beautiful, while $5665$ is not. Determine how many different numbers, when written on the string, make it beautiful.