Let $p(x) = x^3 + ax^2 + bx + c$, where $a$, $b$, and $c$ are complex numbers. Suppose that \[p(2009 + 9002\pi i) = p(2009) = p(9002) = 0\] What is the number of nonreal zeros of $x^{12} + ax^8 + bx^4 + c$?