\[\large \sqrt[8]{527 - \dfrac{1}{527 - \frac{1}{527 - \cdots}}} = \dfrac{\sqrt{a} + \sqrt{b}}{c} , \]

where \(a,b,c\) are positive, square-free integers. Find the product \(abc\).

**Details and Assumptions**:

- To clarify, it is the \(8^\text{th} \) root that is being taken.

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