3. In the Cartesian coordinate system, if a line is at a distance of 1 from point $A(2,2)$, and at a distance of 3 from point $B(m, 0)$, then the set of real values for $m$ for which exactly three such lines exist is $\qquad$.
Gather $\{2-2 \sqrt{3}, 2+2 \sqrt{3}\}$.
Analysis When the circle with center $A$ and radius $r=1$ is externally tangent to the circle with center $B$ and radius $r=3$, there are exactly three common tangents. Using $A B=R+r$ we get $m \in\{2-2 \sqrt{3}, 2+2 \sqrt{3}\}$.