$1$
21. Given trapezoid $A B C D$ with the product of the upper base $A B$ and the height being $\sqrt{2}+1, P$ is a moving point on the lower base $C D$, and line $P A$ intersects the diagonal $B D$ at point $M$. The sum of the areas of $\triangle A M B$ and $\triangle P M D$ is denoted as $S$. Then the minimum value of $S$ is . $\qquad$