Find value range of $2^x+2^y$ Assume $x,y \in \Bbb{R}$ satisfy $$4^x+4^y = 2^{x+1} + 2^{y+1}$$, Find the value range of $$2^x+2^y$$

I know $x=y=1$ is a solution of $4^x+4^y = 2^{x+1} + 2^{y+1}$ , but I can't go further more. I can only find one solution pair of $4^x+4^y = 2^{x+1} + 2^{y+1}$. It seems very far from solve this question... 

Cauchy-Schwarz inequality allows us to conclude $$((a-1)^{2}+(b-1)^{2})(1+1) \geq (a-1+b-1)^{2}$$ and hence $4\geq (a+b-2)^{2}$ and finally $a+b \leq 4$ with equality holding only when $a=b=2$.
