Let $\mathbb{R}$ denote the set of all real numbers. Let $a_i, b_i \in \mathbb{R}$ for $i \in\{1,2,3\}$. Define the functions $f: \mathbb{R} \rightarrow \mathbb{R}, g: \mathbb{R} \rightarrow \mathbb{R}$, and $h: \mathbb{R} \rightarrow \mathbb{R}$ by $$ \begin{aligned} & f(x)=a_1+10 x+a_2 x^2+a_3 x^3+x^4 \\ & g(x)=b_1+3 x+b_2 x^2+b_3 x^3+x^4 \\ & h(x)=f(x+1)-g(x+2) \end{aligned} $$ If $f(x) \neq g(x)$ for every $x \in \mathbb{R}$, then the coefficient of $x^3$ in $h(x)$ is