Consider the polynomial $P(x)$ defined as shown in the image, where $P(x)={\prod }_{k=1}^{10}(x^{2^k}+2^k)=(x^2+2)(x^4+4)(x^8+8)\cdots (x^{1024}+1024)$. The coefficient of $x^{2000}$ in the expansion of $P(x)$ is equal to $2^a$. What is $a$?