BALTIC-WAY 2018 Problem 20

Find all the triples of positive integers $(a, b, c)$ for which the number

$$\frac{(a+b)^{4}}{c}+\frac{(b+c)^{4}}{a}+\frac{(c+a)^{4}}{b}$$

is an integer and $a+b+c$ is a prime.