The Red and Blue circles share the same center.

The radius of Red circle equals \[\cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \cfrac{1}{2 + \cfrac{1}{1 + \ddots}}}}}\] and the radius of Blue circle equals \[3 - \frac{1}{4} + \frac{1\cdot3}{4\cdot6} - \frac{1\cdot3\cdot5}{4\cdot6\cdot8} + \cdots\] If the hatched area equals \[\bigg(\frac{L}{M\sqrt{N}}-1\bigg)\pi+M\] where \(L,M,N\) are positive co-prime integers with \(N\) being square-free. Find \(L+M+N\).

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