Different ways of operating an infinite continued fraction

Question

Given the continued fraction below, $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}+\cfrac{1}{\cfrac{1}{\cdots}+\cfrac{1}{\cdots}}} $$ I wanted to know to which number it converged, so I formulated and solved presumably without any mistake the following equation: $$ x=\frac{1}{x+x}\\ \vdots\\ x=\sqrt{\frac12} $$ After that, I tried to formulate the equation in a different way to see what happened and so I did. But for some reason I couldn't find one solution although I think it's also correctly formulated. $$ x=\frac{1}{\cfrac{1}{x+x}+\cfrac{1}{x+x}}\\ \vdots\\ 0x=0 $$ So, as you can see, there are infinitely many solutions. And this same phenomenum keeps happening if we move the $x$ with which I mean that we could find a solution for this expression: $$ \cfrac{1}{\cfrac{1}{\cfrac{1}{x+x}+\cfrac{1}{x+x}}+\cfrac{1}{\cfrac{1}{x+x}+\cfrac{1}{x+x}}} $$ But not for this one: $$ x=\frac{1}{\cfrac{1}{\cfrac{1}{\cfrac{1}{x+x}+\cfrac{1}{x+x}}+\cfrac{1}{\cfrac{1}{x+x}+\cfrac{1}{x+x}}}+\cfrac{1}{\cfrac{1}{\cfrac{1}{x+x}+\cfrac{1}{x+x}}+\cfrac{1}{\cfrac{1}{x+x}+\cfrac{1}{x+x}}}}\\ $$ My question is: why does that happen? Thanks!