BALTIC-WAY 2016 Problem 4

Let $n$ be a positive integer and let $a, b, c, d$ be integers such that $n \mid a+b+c+d$ and $n \mid a^{2}+b^{2}+c^{2}+d^{2}$. Show that

$$n \mid a^{4}+b^{4}+c^{4}+d^{4}+4 a b c d .$$