2026 IMC keystage3-IWYMIC individual Question 3

Expand and combine like terms in $(-x_1+x_2+x_3+x_4)^4+(x_1-x_2+x_3+x_4)^4+(x_1+x_2-x_3+x_4)^4+(x_1+x_2+x_3-x_4)^4$. What is the smallest coefficient?