2019 AMC 12B 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] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.