Example 6 Let $f(x)=\frac{4^{x}}{4^{x}+2}$. Then, $\sum_{k=1}^{1000} f\left(\frac{k}{1001}\right)$ equals $\qquad$ -
Sampling: From the characteristics of the variable's values, we can deduce that $f(x)$ satisfies the following symmetric property:
If $x+y=1$, then $f(x)+f(y)=1$. Therefore, $f\left(\frac{k}{1001}\right)+f\left(\frac{1001-k}{1001}\right)=1$,
$(k=1,2, \cdots, 1000)$
$$
\therefore \sum_{k=1}^{1000} f\left(\frac{k}{1001}\right)=500 .
$$