3. In the Cartesian coordinate system, $\vec{e}$ is a unit vector, and vector $\vec{a}$ satisfies $\vec{a} \cdot \vec{e}=2$, and $|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$ for any real number $t$, then the range of $|\vec{a}|$ is $\qquad$ .
Let $|\vec{a}|=x$, then $|\vec{a}|^{4} \leqslant 25|\vec{a}+t \vec{e}|^{2} \Rightarrow x^{4} \leqslant 25\left(x^{2}+t^{2}+4 t\right)$ $\Rightarrow 25 t^{2}+100 t+25 x^{2}-x^{4} \geqslant 0$ holds for any real number $t$, thus $\Delta=100^{2}-4 \times 25\left(25 x^{2}-x^{4}\right) \leqslant 0 \Rightarrow x^{4}-25 x^{2}+100 \leqslant 0$ $\Rightarrow 5 \leqslant x^{2} \leqslant 20 \Rightarrow \sqrt{5} \leqslant x \leqslant 2 \sqrt{5}$, so the range of $|\vec{a}|$ is $[\sqrt{5}, 2 \sqrt{5}]$.