A dynamical polynomial is a monicleading coefficient is $1$ polynomial $f(x)$ with integer coefficients such that $f(x)$ divides $f(x^2-2)$.

For example, $f(x)=x^2-x-2$ is a dynamical polynomial because $f(x^2-2)=x^4-5x^2+4=(x^2+x-2)f(x)$.

Let $S(n)$ be the number of dynamical polynomials of degree $n$.
For example, $S(2)=6$, as there are six dynamical polynomials of degree $2$:

$$x^2-4x+4,x^2-x-2,x^2-4,x^2-1,x^2+x-1,x^2+2x+1$$

Also, $S(5)=58$ and $S(20)=122087$.

Find $S(10000)$. Give your answer modulo $998244353$.