3. In the Cartesian coordinate system, $\boldsymbol{e}$ is a unit vector, and vector $\boldsymbol{a}$ satisfies $\boldsymbol{a} \cdot \boldsymbol{e}=2$, and $|\boldsymbol{a}|^{2} \leqslant 5|\boldsymbol{a}+t \boldsymbol{e}|$ for any real number $t$. Then the range of $|\boldsymbol{a}|$ is $\qquad$ .
3. $[\sqrt{5}, 2 \sqrt{5}]$.

Let $\boldsymbol{e}=(1,0)$.
From $\boldsymbol{a} \cdot \boldsymbol{e}=2$, we can set $\boldsymbol{a}=(2, s)$, then for any
real number $t$, we have
$$
4+s^{2}=|\boldsymbol{a}|^{2} \leqslant 5|\boldsymbol{a}+\boldsymbol{t} \boldsymbol{e}|=5 \sqrt{(2+t)^{2}+s^{2}},
$$

which is equivalent to $4+s^{2} \leqslant 5|s|$, solving this gives $|s| \in[1,4]$.
Thus $s^{2} \in[1,16]$
$$
\Rightarrow|\boldsymbol{a}|=\sqrt{4+s^{2}} \in[\sqrt{5}, 2 \sqrt{5}] .
$$