Define a sequence recursively by $x_0=4$ and $x_{n+1}=\frac{x_n^2+4x_n+3}{x_n+5}$ for all nonnegative integers $n$. Let $m$ be the least positive integer such that $x_m\le 3+\frac{1}{2^{20}}$. In which of the following intervals does $m$ lie?