3. On the number line, points with coordinates $1, 2, \cdots, 2006$ are called "marked points". A frog starts from point 1, and after 2006 jumps, it visits all the marked points and returns to the starting point. What is the maximum length of the path the frog can jump? Explain your reasoning.
(2006, Shandong Province Junior High School Mathematics Competition)
Let the points the frog reaches in sequence be $x_{1}, x_{2}, \cdots, x_{2006}\left(x_{1}=1\right)$. Then the total length of the jumps is
$$
S=\left|x_{1}-x_{2}\right|+\left|x_{2}-x_{3}\right|+\cdots+\left|x_{2006}-x_{1}\right| \text {. }
$$

We call $\{1,2, \cdots, 1003\}$ the upper half, and $\{1004, 1005, \cdots, 2006\}$ the lower half. If we make the points in the upper half take the “-” sign twice during the calculation, and the points in the lower half take the “+” sign twice, then
$$
\begin{aligned}
S_{\text {Ao Da }}= & 2(1004+1005+\cdots+2006)- \\
& 2(1+2+\cdots+1003) \\
= & 2 \times 1003^{2} .
\end{aligned}
$$

In fact, as long as each jump is from the upper half to the lower half, or from the lower half to the upper half, the points in the upper half will take “-”, and the points in the lower half will take “+”.