Example 2 Figure 1 is a map of an industrial area, a highway (thick line) runs through the area, and seven factories $A_{1}, A_{2}$, $\cdots, A_{7}$ are distributed on both sides of the highway, connected to the highway by some small roads (thin lines). Now, a long-distance bus station is to be set up on the highway, with the goal of minimizing the total distance from the station to each factory (along the highway and small roads).
Question: (1) Where is the best place to set up this station?
(2) Prove your conclusion.
(3) If another factory is built at point $P$ and a small road is constructed along the dashed line in the figure, where should the station be set up now?
【Analysis】The optimal station location is the point that minimizes the sum of the distances from the seven factories to the station.

Since the distances from each factory to the road intersection are fixed, we only need to consider the sum of the distances from the five small intersections $F, E, D, C, B$ to the station. This can be transformed into a mathematical problem.

Let the distances from points $E, D, C, B$ to point $F$ be $a, b, c, d$ respectively. Clearly, the curvature of the road does not affect the problem, so we can "straighten" the road and establish a number line with $F$ as the origin. Let the station position be point $M$, and the corresponding real number of $M$ be $x$. Then the problem is to find the value of $x$ when the function
$$
\begin{aligned}
f(x)= & 2|x|+|x-a|+2|x-b|+ \\
& |x-c|+|x-d|
\end{aligned}
$$
achieves its minimum value.
By breaking down the absolute value expressions in the function $f(x)$, making all coefficients 1, and arranging them in ascending order of the absolute value zeros, we get
$$
\begin{aligned}
f(x)= & |x|+|x|+|x-a|+|x-b|+ \\
& |x-b|+|x-c|+|x-d| .
\end{aligned}
$$

In the above sum, there are seven terms, with the middle one being $|x-b|$. Therefore, by Conclusion 1, we know that when $x=b$, $f(x)$ achieves its minimum value.
Thus, setting the station at $D$ is the most reasonable.
If another factory is built at point $P$, then the problem is to find the minimum value point of the function
$$
\begin{array}{c}
g(x)=3|x|+|x-a|+2|x-b|+ \\
|x-c|+|x-d| \\
=|x|+|x|+|x|+|x-a|+|x-b|+ \\
|x-b|+|x-c|+|x-d|
\end{array}
$$

By Conclusion 1, we know that the minimum value point of the function $g(x)$ is any number in the interval $[a, b]$.
Therefore, setting the station between points $E$ and $D$ is the best.