Consider the two-dimensional vector field $\vec{F}(x,y)=x\,\hat{i}+y\,\hat{j}$, where $\hat{i}$ and $\hat{j}$ denote the unit vectors along the $x$-axis and the $y$-axis, respectively. A contour $C$ in the $xy$-plane, as shown in the figure, is composed of two horizontal lines connected at the two ends by two semicircular arcs of unit radius. The contour is traversed in the counter-clockwise sense. The value of the closed path integral
\[
\oint_{C} \vec{F}(x,y)\cdot(dx\,\hat{i}+dy\,\hat{j})
\]
is ________________.
