Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function such that its derivative $f'$ is continuous and $f(\pi) = -6$.
If $F: [0, \pi] \rightarrow \mathbb{R}$ is defined by $F(x) = \int_{0}^{x} f(t)\, dt$, and if
$$\int_{0}^{\pi} \left(f'(x) + F(x)\right) \cos x\, dx = 2$$
then the value of $f(0)$ is $\_\_\_\_$