Nested Impossibilities

Algebra Level 4

\[4+\frac{1}{2+\frac{1}{1+\frac{1}{3+\frac{1}{1+\frac{1}{2+\frac{1}{8+\frac{1}{\ldots} } } } } } } = \sqrt{ A} \]

Find the positive integer \(A\) in the equation above.

Details and Assumptions: The pattern repeats \(2,1,3,1,2,8 \) infinitely, but the \(4\) comes only one time, i.e. in the beginning.

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