Alexander has chosen a natural number $N > 1$ and has written down, in increasing order, all its positive divisors $d_1 < d_2 < \ldots < d_s$ (where $d_1 = 1$ and $d_s = N$). For each pair of neighboring numbers, he has found their greatest common divisor. The sum of all these $s-1$ numbers (the greatest common divisors) is equal to $N - 2$. Find all possible values of $N$.

Let's think step by step and output the final answer within \boxed{}.
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