Can't see what Ramanujan did there

Number Theory Level 5

If the infinite continued fraction \[ \cfrac{1}{1+\cfrac{e^{-2\pi}}{1+\cfrac{e^{-4\pi}}{1+\cdots}}}\] can be expressed as \[e^{\dfrac{a\pi}{b}}\left(\sqrt{\dfrac{b+\sqrt{b}}{a}}-\dfrac{c+\sqrt{b}}{a}\right)\] where \(a,b\) and \(c\) are coprime integers,find \(a+b+c\).

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