Solve $1-4^{x}+2^{x+1}=2^{x}+2^{-x}$.
Let $t=2^{x}$. The equation is equivalent to $1-t^{2}+2 t=t+1 / t$, which simplifies to $1-t^{2}+t-1 / t=0$. By multiplying by $t$, we get $t(1-t^{2})+t^{2}-1=0$, or equivalently $0=(1-t^{2})(t-1)=(1-t)(1+t)(t-1)=-(t+1)(t-1)^{2}$.

Therefore, $t=1$ or $t=-1$. Since $t=2^{x}>0$, we necessarily have $t=1$, hence $x=0$.

Conversely, we verify that $x=0$ is indeed a solution.