Fraction

Calculus Level 3

\[ \large 1 + \cfrac{1}{2 + \cfrac{1}{1+\cfrac{1}{2+\cfrac{1}{1 + \ddots}}}} \]

If the infinitely nested fraction above is equal to \( \dfrac{a+\sqrt b}c \), where \(a,b\) and \(c\) are positive integers with \(b\) square-free, find \(a+b+c\).

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