New Year's Countdown Day 30: Goodbye 2017

Algebra Level 2

\[ \large { \sqrt[30]{20x +\sqrt[30]{20x + \sqrt[30]{20x + 17}}} = 17 } \]

Let \(N\) be the sum of all the real solutions to the above equation. If \(N = \dfrac{a^b-a}{c},\) where \(a,\) \(b,\) and \(c\) are positive integers and \(a\) and \(c\) are coprime, then what is \(a+b+c?\)

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