Given positive integers $m_1,...,m_n$, is it possible to solve the following equation system over the field of complex numbers?

$$m_1x_1+\cdots+m_nx_n=0$$ $$m_1x_1^2+\cdots+m_nx_n^2=0$$ $$\cdots$$ $$m_1x_1^{n-1}+\cdots+m_nx_n^{n-1}=0$$ $$x_1x_2\cdots x_n=1.$$

When $n=1,2,3,4$, I can find a formula for the solutions. But I can not do it for $n>4$. Also, when $m_1=m_2=...=m_n=1$, the solutions can be written as

$$(\zeta^{\sigma(1)},\cdots,\zeta^{\sigma(n)})$$

where $\zeta$ is a $n$-th primitive root of unity and $\sigma$ is an element of the symmetric group $S_n$.

For other concrete examples, the Mathematica numerical computation shows that the number of solutions to this equation system would be $n!$. Does anybody have an idea to prove it?

In general, if we do not assume $m_1,...,m_n$ are positive integers, what is the condition on $m_1,...,m_n$ such that this equation system has a solution?