Example 3 Let the even function $f(x)$ and the odd function $g(x)$ both be defined on $[-3,3]$, and satisfy
$$
f(x)+g(x) \geqslant 2007 x \sqrt{9-x^{2}}+x^{2006}
$$

Find $f(x)$ and $g(x)$.
Given the function $p(x)(\forall p(x) \geqslant 0, x \in[-3,3])$, let
$$
f(x)+g(x)=2007 x \sqrt{9-x^{2}}+x^{2008}+p(x)
$$

Substitute $-x$ for $x$ in the above equation, we get
$$
f(x)-g(x)=-2007 x \sqrt{9-x^{2}}+x^{2008}+p(-x)
$$

Solving the system of equations (1) and (2), we obtain the solution to the functional inequality as
$$
\begin{array}{c}
f(x)=x^{2000}+\frac{p(x)+p(-x)}{2}, x \in[-3,3] \\
g(x)=2007 x \sqrt{9-x^{2}}+\frac{p(x)-p(-x)}{2}, x \in[-3,3]
\end{array}
$$

where the parameter function $\forall p(x) \geqslant 0, x \in[-3,3]$.
Note: Since $\frac{p(x)+p(-x)}{2}$ is an even function and $\frac{p(x)-p(-x)}{2}$ is an odd function, the conclusion of this problem cannot be rewritten in a simpler form using inequalities as in Example 1 and Example 2.