9. Given real numbers $x, y$ satisfy $2^{x}+3^{y}=4^{x}+9^{y}$, try to find the range of $U=8^{x}+27^{y}$.
Let $u=2^{x}, v=3^{y}$, then $u+v=u^{2}+v^{2} \Rightarrow (u+v)^{2}=u+v+2uv \geqslant 4uv$, $\Rightarrow 0<2uv=(u+v)^{2}-(u+v) \leqslant u+v \Rightarrow 1<u+v \leqslant 2$.
Thus, $u^{3}+v^{3}=(u+v)^{3}-3uv(u+v)=\frac{3}{2}(u+v)^{2}-\frac{1}{2}(u+v)^{3}$, let $t=u+v \in(1,2], y=\frac{3}{2}t^{2}-\frac{1}{2}t^{3}$, then $y'=3t-\frac{3}{2}t^{2}=3t\left(1-\frac{1}{2}t\right) \geqslant 0 \Rightarrow y$ is monotonically increasing, so $y \in(1,2] \Rightarrow U \in(1,2]$.