2021 Fall AMC 12B Problem 18

Set u_0 = \frac{1}{4}, and for k \ge 0 let u_{k+1} be determined by the recurrence u_{k+1} = 2u_k - 2u_k^2.

This sequence tends to a limit; call it L. What is the least value of k such that |u_k-L| \le \frac{1}{2^{1000}}?

\textbf{(A)}\: 10\qquad\textbf{(B)}\: 87\qquad\textbf{(C)}\: 123\qquad\textbf{(D)}\: 329\qquad\textbf{(E)}\: 401

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