I'm going to try to answer my own question.
Basically, when you try to graph the inequality of $|\min(X,Y)|<1$, you will get a L-shape graph. And the area of the function can be calculated as the following
$\begin{align} \operatorname{Area}(|\min(X,Y)|<1) &= \operatorname{Area}(-1-1) + \operatorname{Area}(-11)\\ &=\operatorname{Area}(-1-1) + \operatorname{Area}(-11)\\ &=\operatorname{Area}(-1
It is like rotating the lower right piece of that L-shape graph 90 degrees clockwise.
Then the probability of $P(|\min(X,Y)|<1)$ can be easily calculated.
$P(|\min(X,Y)|<1) = P(-1