Solve the following system in real numbers: \begin{cases} \log_2(x+y)+4=2^x+2^y \\ \frac{x+y}{4}+\frac{xy}{x+y}=1 \end{cases}
I used the fact that $\frac{xy}{x+y} \leq \frac{x+y}{4}$ in order to get $x+y \geq 2$, but I couldn't find even one solution afterwards.