Let $S:=\{(a, b) \mid 0 \leq a \leq 1,0 \leq b \leq 1\}$, a unit square, in $\mathbb{R}^{2}$. Let $B:=$ $\left\{(x, y) \mid x^{2}+y^{2} \leq 1\right\}$, a unit disk, in $\mathbb{R}^{2}$. Define the set $S+B$ as follows:
\[
S+B:=\{(u, v) \mid \exists(a, b) \in S,(x, y) \in B \text { such that } u=a+x, v=b+y\} .
\]
What is the area of $S+B$ ?
- $\pi+4$
- $\pi+5$
- $\pi+3$
- $\pi+2$
- None of the above.
