It's all about number theory! (1)

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

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 $$a-d$$.