Let \( \mathbb{N} \) denote the set of all natural numbers, and \( \mathbb{Z} \) denote the set of all integers. Consider the functions \( f:\mathbb{N}\to \mathbb{Z} \) and \( g:\mathbb{Z}\to \mathbb{N} \) defined by 
Define \( (g^\circ f)(n)=g(f(n)) \) for all \( n\in \mathbb{N} \), and \( (f^\circ g)(n)=f(g(n)) \) for all \( n\in \mathbb{Z} \).
Then which of the following statements is(are) TRUE?
