Please help me for solving a homework.
Let $k$ be a field and $R=k[[x_1,x_2,\ldots,x_n]]$ the ring of power series over $k$. If $I$ is an ideal of $R$ such that $(I:x_1)=(I:x_2)=(I:x_3)=\cdots=(I:x_{r})\neq (x_1,x_2,\ldots,x_n)$ and $(I:x_{r+1})=(I:x_{r+2})=\cdots=(I:x_n)=(x_1,x_2,\ldots,x_n)$ for some $r\in\{1,2,\ldots,n \} $, and also $(x_1,x_2,\ldots,x_n)^3 \subseteq I $, then what can we say about the generators of $I$ and the number $n$ (is it one)?