Let $n$ be a positive integer. If $r\hspace{0.25mm} \equiv \hspace{1mm} n\hspace{1mm} (mod\hspace{1mm} 2)$ and $r\hspace{0.10mm} \in \hspace{0.10mm} \{ 0,\hspace{0.10mm} 1 \} $, find the number of integer solutions to the system of equations $\left\{\begin{array}{l}x+y+z = r \\ \mid x \mid + \mid y \mid + \mid z \mid = n \end{array}\right.$