3. In the Cartesian coordinate system $x O y$, two point sets $M, N$ satisfy $M=\left\{(x, y) \mid x^{2}+y^{2}=1\right\}$, $N=\{(x, y)|| x-1|+| y-1 \mid=a\}$. If $M \bigcap N=\varnothing$, then the range of the real number $a$ is $\qquad$
Solution:
As shown in the figure, set $N$ is the square $A B C D$, where $C D: x+y+a-2=0$. Therefore, $d=\frac{|a-2|}{\sqrt{2}}>1$ $\Rightarrow a>2+\sqrt{2}$ or $a<2-\sqrt{2}$.
Thus, the range of the real number $a$ is $(-\infty, 2-\sqrt{2}) \cup(2+\sqrt{2},+\infty)$.