12. In the Cartesian coordinate system $x O y$, the area of the plane region corresponding to the point set $S=\left\{(x, y) \mid(|x|+|y|-1)\left(x^{2}+y^{2}-1\right) \leqslant 0\right\}$ is $\qquad$ .
According to the problem, we have
\[
\left\{\begin{array}{l}
|x|+|y|-1 \geqslant 0, \\
x^{2}+y^{2}-1 \leqslant 0,
\end{array}\right.
\]
The plane region corresponding to $S$ is shown in the shaded area in the figure, and its area is $\pi-2$.