Suppose A is a three-dimensional $a × b × c$ rectangular brick with $a, b, c > 0$. Suppose B is the set of all points whose distance from a point in A is at most 1 (in particular, B contains A), and the volume of B is $a b c+m(a b+a c+b c)+\pi(a+b+c)+\frac{n \pi}{p}$. Find the value of $m+n+p$.

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