Example 3 Let sets $A$ and $B$ be subsets of the universal set $U=\{1,2,3,4\}$. It is known that $(\complement_{U} A) \cap B=\{1\}, A \cap B=\{3\}, (\complement_{U} A) \cap (\complement_{U} B)=\{2\}$. Find $\complement_{U}(A \cup B)$.
Analysis We draw a Venn diagram to intuitively illustrate their relationship.
Solution As shown in Figure 1-1, use a rectangle to represent the universal set $U$, and use two closed curves to represent sets $A$ and $B$. Given $(\complement_{U} A) \cap B=\{1\}$, fill in 1 in the area outside $A$ but inside $B$; given $A \cap B=\{3\}$, fill in 3 in the common area of $A$ and $B$; and given $(\complement_{U} A) \cap (\complement_{U} B)=\{2\}$, fill in 2 in the area outside both $A$ and $B$, but inside the rectangle. Since the universal set $U=\{1,2,3,4\}$, 4 should be filled in the area inside $A$ but outside $B$. From Figure 1-1, we can see that $A \cup B=\{1,3,4\}$, thus $\complement_{U}(A \cup B)=\{2\}$.