AMC 12 2016b Problem 22

For a certain positive integer $n$ less than $1000$, the decimal equivalent of $\frac{1}{n}$ is $0.\overline{abcdef}$, a repeating decimal of period of $6$, and the decimal equivalent of $\frac{1}{n+6}$ is $0.\overline{wxyz}$, a repeating decimal of period $4$. In which interval does $n$ lie?

\textbf{(A)}\ [1,200] \textbf{(B)}\ [201,400] \textbf{(C)}\ [401,600] \textbf{(D)}\ [601,800] \textbf{(E)}\ [801,999]