Let $m$ be the maximum possible value of $x^{16} + \frac{1}{x^{16}}$, where \[x^6 - 4x^4 - 6x^3 - 4x^2 + 1=0.\] Find the remainder when $m$ is divided by $2007$.

Let's think step by step and output the final answer within \boxed{}.
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