Example 16 (from the 2nd American Mathematical Invitational Competition) The function $f(x)$ is defined on the real number domain and satisfies the following conditions: for any real number $x$, $f(2+x)=f(2-x)$, and $f(7+x)=f(7-x)$. If $x=0$ is a root of the equation $f(x)=0$, how many roots should the equation $f(x)=0$ have at least in the interval $-1000 \leqslant x \leqslant 1000$?