Find the two complex-valued functions $F^{+}(z)$ and $F^{-}(z)$ such that all of the following hold:

(i) $F^{+}(z)$ and $F^{-}(z)$ are analytic for $\operatorname{Im} z>0$ and $\operatorname{Im} z<0$ respectively, where $z=x+i y, x, y \in \mathbb{R}$.

(ii) $F^{+}(x)-F^{-}(x)=\frac{1}{x^{4}+1}, \quad x \in \mathbb{R}$.

(iii) $F^{\pm}(z)=O\left(\frac{1}{z}\right), \quad z \rightarrow \infty, \quad \operatorname{Im} z \neq 0$.