Trying to find Green's function for an ODE with Periodic Boundary Conditions?

Question

So I am trying to solve the following ODE: $$-\frac{d^2f(x)}{dx^2}=b(x)$$ For a general $b(x)$ where $f(-L/2)=f(L/2),f'(-L/2)=f'(L/2)$ and $b(-L/2)=b(L/2), b'(-L/2)=b'(L/2)$. The hint the problem gave was to find a Green's function of the form $G(x,y)$ such that: $$f(x)=\int_{-L/2}^{L/2}G(x,y)b(y)dy$$ And so that $G(x,y)$ is periodic in $x$ and $y$. We have used Green's functions before but only for Dirichlet boundary condtions so I'm a little lost about what to do. I wanted to solve the ode: $$-\frac{d^2G(x,y)}{dx^2}=\delta(x-y)$$ And then argue that would satisfy the $f(x)$ relation, but I'm not entirely sure how to do with this with the periodic boundary conditions. Would a Fourier or Laplace transform maybe help here? I'm honestly kind of lost. Any help would be appreciated.