There is a polynomial $P$ such that for every real number $x$,
\[
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).
\]When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?
<|im_start|>system
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<|im_start|>user
There is a polynomial $P$ such that for every real number $x$,
\[
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).
\]When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
341
There is a polynomial $P$ such that for every real number $x$,
\[
  x^{512} + x^{256} + 1 = (x^2 + x + 1) P(x).
\]When $P$ is written in standard polynomial form, how many of its coefficients are nonzero?
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