Set $u_0=\frac{3}{8}$, 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 $|u_k-L|\le \frac{1}{2^{1000}}$?