Let
\[
f(x)=\left\{\begin{array}{cr}
\pi+x, & -\pi \leq x<0, \\
0, & 0 \leq x<\pi,
\end{array}\right.
\]
with $f(x+2 \pi)=f(x)$. If $F(x)$ represents the Fourier series of $f(x)$, then the value of $F\left(-\frac{\pi}{2}\right)+F(0)$ is
- $0$
- $\frac{\pi}{2}$
- $\pi$
- $\frac{3 \pi}{2}$
