Maximizing the Sum of Pairs Satisfying an Equation

Consider all positive integer solutions (xi,yi)(x_i,y_i) to

1+2x+22x+1=y21+2^x+2^{2x+1}=y^2

Find the maximum value of the sum xi+yix_i+y_i.


This problem is based off of an old IMO problem. Good luck!
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