Example 5 If $a, c, d$ are integers, $b$ is a positive integer, and they satisfy $a+b=c, b+c=d, c+d=a$, then the maximum value of $a+b+c+d$ is ( ).
(A) -1
(B) -5
(C) 0
(D) 1
(1991, National Junior High School Competition)
Analysis: Since $b$ is a positive integer, we can consider $b$ as the main variable and try to eliminate $a, c, d$.

Solution: From the given, we have $c-a=b, d-c=b, c+d-a=0$. Solving the system of equations for $a, c, d$ yields
$$
a=-3b, c=-2b, d=-b \text{. }
$$

Thus, $a+b+c+d=-5b \leqslant-5$. (Since $b$ is a positive integer)

Therefore, the minimum value of $a+b+c+d$ is -5. The correct choice is (B).