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 $L$. Find the least value of $k$ such that 
$$|u_k-L| \le \frac{1}{2^{1000}}.$$

Let's think step by step and output the final answer within \boxed{}.
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