The largest solution to
\[9x^3 - 20x = 8 \sqrt{2}\]can be written in the form $\frac{\sqrt{a} + \sqrt{b}}{c},$ where $a,$ $b,$ and $c$ are positive integers, when simplified.  Find $a + b + c.$

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