Four, (18 points) The real numbers $x_{1}, x_{2}, \cdots, x_{2001}$ satisfy
$$
\begin{array}{l}
\sum_{k=1}^{2000}\left|x_{k}-x_{k+1}\right|=2001 . \\
\text { Let } y_{k}=\frac{1}{k}\left(x_{1}+x_{2}+\cdots+x_{k}\right), k=1,2 .
\end{array}
$$
$\cdots, 2$ 001. Find the maximum possible value of $\sum_{k=1}^{2000}\left|y_{k}-y_{k+1}\right|$.