AMC 12 2019b Problem 22

Define a sequence recursively by $x_0=5$ and $$x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}$$for all nonnegative integers $n.$ Let $m$ be the least positive integer such that $$x_m\leq 4+\frac{1}{2^{20}}.$$In which of the following intervals does $m$ lie?

$\textbf{(A) } [9,26] \textbf{(B) } [27,80] \textbf{(C) } [81,242] \textbf{(D) } [243,728] \textbf{(E) } [729,\infty]$