3. Inside a cube $C$ with edge length 1, construct a large inscribed sphere $O_{1}$, and then construct a smaller sphere $O_{2}$ inside $C$ such that it is externally tangent to the large sphere $O_{1}$ and also tangent to three faces of the cube. The surface area of sphere $O_{2}$ is $\qquad$.
Given the sphere $O_{1}$ with radius $R=\frac{1}{2} \Rightarrow C D=\frac{\sqrt{3}-1}{2}$, by the projection theorem we get $C D^{2}=N D \cdot D E=2 D E^{2}$ $\Rightarrow C D=\sqrt{2} D E, C E=\sqrt{3} D E$. In $\triangle C D E$, let $O_{2} F=O_{2} D=r, \frac{r}{D E}=\frac{C D-r}{C E}$ $\Rightarrow r=\frac{C D}{\sqrt{3}+1}=\left(\frac{\sqrt{3}-1}{2}\right)^{2}=\frac{2-\sqrt{3}}{2}$, so the surface area of sphere $O_{2}$ is $(7-4 \sqrt{3}) \pi$.