Four positive integers $a$, $b$, $c$, and $d$ have a product of $8!$ and satisfy:  
  
$$\begin{array}{rl} ab + a + b & = 524 \\ bc + b + c & = 146 \\ cd + c + d & = 104 \end{array}$$  
  
Find the value of $a-d$.

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