Let $n \geq 100$ be a positive integer. Let $X_{1}, X_{2}, \ldots, X_{n}$ be independent random variables, each taking values in the set $\{0,1\}$ such that $\operatorname{Pr}\left[X_{i}=1\right]=\frac{2}{3}$ for each $1 \leq i \leq n$. Define $S:=\sum_{i=1}^{n} X_{i}$, and let $p(x)$ be a polynomial such that for each non-negative integer $j$, the coefficient of $x^{j}$ in $p(x)$ is $\operatorname{Pr}[S=j]$. What is $p^{\prime}(1)$ ?
Note: $p^{\prime}(x)$ denotes the polynomial obtained by differentiating $p$ with respect to $x$.
- $0$
- $n$
- $\frac{2 n}{3}$
- $\frac{4 n^{2}+2 n}{9}$
- $\frac{4 n^{2}-4 n}{9}$
