Example 5 Find the value of $2-2^{2}-2^{3}-\cdots-2^{2005}+2^{2000}$.
Explanation: Utilizing the characteristic $2^{n+1}-2^{n}=2^{n}$, rearrange the terms of the original expression in reverse order.
Therefore, the original expression
$$
\begin{array}{l}
=2^{2006}-2^{2005}-2^{2004}-\cdots-2^{3}-2^{2}+2 \\
=2^{2006}(2-1)-2^{2005}-\cdots-2^{3}-2^{2}+2 \\
=2^{2005}(2-1)-2^{2004}-\cdots-2^{3}-2^{2}+2 \\
=\cdots \\
=2^{2}(2-1)+2=6 .
\end{array}
$$