Let $n$ be a positive integer and consider the system \begin{align*} S(n):\begin{cases} x^2+ny^2=z^2\\ nx^2+y^2=t^2 \end{cases}, \end{align*}where $x,y,z$, and $t$ are naturals. If $M_1=\{n\in\mathbb N:$ system $S(n)$ has infinitely many solutions$\}$, and $M_1=\{n\in\mathbb N:$ system $S(n)$ has no solutions$\}$, prove that $7 \in M_1$ and $10 \in M_2$. sets $M_1$ and $M_2$ are infinite.