HMMT 2015 A10

Find all ordered 4 -tuples of integers $(a, b, c, d)$ (not necessarily distinct) satisfying the following system of equations: $$$$ \begin{aligned} a^{2}-b^{2}-c^{2}-d^{2} & =c-b-2 \ 2 a b & =a-d-32 \ 2 a c & =28-a-d \ 2 a d & =b+c+31 \end{aligned} $$$$