Consider the polynomial $P(x)$ defined as shown in the image, where $P(x)={\prod }_{k=0}^9(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{512}+512)$. The coefficient of $x^{1000}$ in the expansion of $P(x)$ is equal to $2^a$. What is $a$?