A logarithmic equation

Algebra Level 4

The sum of all (positive) solutions of the equation

$\log_{16}x +\log _x 16=\log_{512} x + \log_x {512}$

can be written as $\frac{a}{b}$ , where $a$ and $b$ are coprime positive integers. What are the last three digits of $a+b$ ?

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