2021 AIME II Question 3

Find the number of permutations $x_1,$ $x_2,$ $x_3,$ $x_4,$ $x_5$ of numbers $1,$ $2,$ $3,$ $4,$ $5$ such that the sum of five products $$
\begin{aligned} &x_1x_2x_3 + x_2x_3x_4 \\ &\quad {}+ x_3x_4x_5 + x_4x_5x_1 \\ &\quad {}+ x_5x_1x_2 \end{aligned}
$$ is divisible by $3.$