Nested Impossibilities

Algebra Level 3

$\Large 4+\frac{1}{2+\frac{1}{1+\frac{1}{3+\frac{1}{1+\frac{1}{2+\frac{1}{8+\frac{1}{\ddots} } } } } } } = \sqrt{ A}$

Find the positive integer $A$ 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.