Four positive integers a, b, c, and d have a product of 8 ! and satisfy$$

\begin{aligned}

a b+a+b=524 \

b c+b+c=146, \text { and } \

c d+c+d=104

\end{aligned}

What is $a-d$ ?
**Answer Choices**
A. $4$
B. $6$
C. $8$
D. $10$
E. $12$

Four positive integers a, b, c, and d have a product of 8 ! and satisfy$$

\begin{aligned}

a b+a+b=524 \

b c+b+c=146, \text { and } \

c d+c+d=104

\end{aligned}

What is $a-d$ ?
**Answer Choices**
A. $4$
B. $6$
C. $8$
D. $10$
E. $12$