Let $\mathbb{N}$ denote the set of all natural numbers, and $\mathbb{Z}$ denote the set of all integers. Consider the functions $f: \mathbb{N} \rightarrow \mathbb{Z}$ and $g: \mathbb{Z} \rightarrow \mathbb{N}$ defined by
$ f(n)= \begin{cases}(n+1) / 2 & \text { if } n \text { is odd } \\ (4-n) / 2 & \text { if } n \text { is even }\end{cases} $
and $g(n)= \begin{cases}3+2 n & \text { if } n \geq 0 \\ -2 n & \text { if } n<0\end{cases}$
Define $\left(g^{\circ} f\right)(n)=g(f(n))$ for all $n \in \mathbb{N}$, and $\left(f^{\circ} g\right)(n)=f(g(n))$ for all $n \in \mathbb{Z}$.
Then which of the following statements is (are) TRUE?