Set $u{0}=\frac{1}{4}$, and for $k \geq 0$ let $u{k+1}$ be determined by the recurrence
$u{k+1}=2 u{k}-2 u{k}^{2}$
This sequence tends to a limit; call it $L$. What is the least value of $k$ such that
$|u{k}-L| \leq \frac{1}{2^{1000}} ?$