1. Let $X=\{1,2, \cdots, 2001\}$. Find the smallest positive integer $m$, such that: for any $m$-element subset $W$ of $X$, there exist $u, v \in W$ (where $u, v$ can be the same), such that $u+v$ is a power of 2.
Hint: Construct drawers such that the sum of two numbers in the drawers is a power of 2, some drawers contain only one number that is a power of 2. Answer: 999.