3. As shown in Figure $10, A B C D$ is a square with a side length of 1, $U$ and $V$ are points on $A B$ and $C D$ respectively, $A V$ intersects $D U$ at point $P$, and $B V$ intersects $C U$ at point $Q$. Find the maximum value of the area of quadrilateral $P U Q V$.
(2000, Shanghai Junior High School Mathematics Competition)
(Hint: Use the method of extreme by transforming the formula. Answer: $\frac{1}{4}$. )