It's all about number theory! (1)

Number Theory Level 4

\[\begin{cases} ab+a+b = 524\\ bc+b+c = 146 \\ cd+c+d = 104 \end{cases} \]

The product of four positive integers \(a\), \(b\), \(c\), and \(d\) is \(8! \). If these numbers satisfy the system of equations above, find the value of \(a-d\).

See also It's all about number theory! (2).
This problem is not original. It is taken from one of the problems at a local olympiad in Medellín, Colombia.

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