Difficult system of equations

Algebra Level 4

\[ \large { \begin{cases} a + \frac{a+8b}{a^2+b^2} = 2 \\ b + \frac{8a-b}{a^2+b^2} = 0 \end{cases} } \]

If \((a_1,b_1), (a_2,b_2) , \ldots ,(a_n,b_n) \) are all the real solutions of \((a,b) \) which satisfy the system of equations above, find the value of \(\displaystyle \prod_{m=1}^n a_m b_m \).

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