IL-GILLIS 2020 Problem 4

At the start of the day, the four numbers $(a_0,b_0,c_0,d_0)$ were written on the board. Every minute, Danny replaces the four numbers written on the board with new ones according to the following rule: if the numbers written on the board are $(a,b,c,d)$ , then Danny first calculates the numbers \begin{align*} a'&=a+4b+16c+64d b'&=b+4c+16d+64a c'&=c+4d+16a+64b d'&=d+4a+16b+64c \end{align*} and replaces the numbers $(a,b,c,d)$ with the numbers $(a'd',c'd',c'b',b'a')$ . For which initial quadruples $(a_0,b_0,c_0,d_0)$ , will Danny write at some point a quadruple of numbers all of which are divisible by $5780^{5780}$ ?