# Number of Terms in Expansion of Polynomial

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Let

\begin{align*}P(x)&=\displaystyle\prod_{n=1}^{10}\left(x^{2^n}-x^{2^{n-1}}+1\right)\\&=(x^2-x+1)(x^4-x^2+1)\ldots(x^{1024}-x^{512}+1).\end{align*}

Let $$N$$ be the number of terms with non-zero coefficients when $$P(x)$$ is expanded (including the constant term). What is the remainder when $$N$$ is divided by $$1000$$?

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