BALTIC-WAY 2016 Problem 9

Find all quadruples $(a, b, c, d)$ of real numbers that simultaneously satisfy the following equations:

$$\left\{\begin{aligned} a^{3}+c^{3} & =2 \\ a^{2} b+c^{2} d & =0 \\ b^{3}+d^{3} & =1 \\ a b^{2}+c d^{2} & =-6 \end{aligned}\right.$$