Nested Impossibilities

Algebra Level 3

4+12+11+13+11+12+18+1=A\Large 4+\frac{1}{2+\frac{1}{1+\frac{1}{3+\frac{1}{1+\frac{1}{2+\frac{1}{8+\frac{1}{\ddots} } } } } } } = \sqrt{ A}

Find the positive integer AA in the equation above.


Details and Assumptions:

  • The pattern repeats 2,1,3,1,2,82,1,3,1,2,8 infinitely, but the 44 comes only one time, i.e. in the beginning.
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