Mock AIME 3 Pre 2005 Problems/Problem 8

Problem

Let $N$ denote the number of $8$ -tuples $(a_1, a_2, \dots, a_8)$ of real numbers such that $a_1 = 10$ and

$\left|a_1^{2} - a_2^{2}\right| = 10$

$\left|a_2^{2} - a_3^{2}\right| = 20$

$\cdots$

$\left|a_7^{2} - a_8^{2}\right| = 70$

$\left|a_8^{2} - a_1^{2}\right| = 80$

Determine the remainder obtained when $N$ is divided by $1000$ .

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