1, 5, 2, and 3

\((1,5,2,3)\) are 4 distinct positive integers such that the sum of 2 of them equals the product of the other 2, and vice versa, i.e. the sum of these 2 other numbers equals the product of the 2 numbers initially mentioned: \[1+5=2\times 3, \quad 2+3=1\times 5.\] Let it be generalized with \(a, b, c, d\) \[a+b=c\times d, \quad c+d=a\times b.\] Determine the number of un-ordered quadruplets \((a,b,c,d) \) other than \((1,5,2,3) \) that satisfy this condition.

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