KöMaL 2018 B4968

Solve the following simultaneous equations on the set of positive real numbers: $$\begin{align*} \frac{1}{1+a + ab + abc} + \frac{1}{1+b + bc + bcd} + \frac{1}{1+c + cd + cda} + \frac{1}{1+d + da + dab} & = 1,\\ a+b+c+d & = 4. \end{align*}$$