Verify that the following numbers are constructible by determining the sequence of their extension fields:
$$ \sqrt{1+\sqrt2+\sqrt3+\sqrt5}, \frac{\sqrt5+\sqrt{11}}{1+\sqrt{7-\sqrt3}},\frac{\sqrt{2+\sqrt3}}{\sqrt[3]{2}+\sqrt{1+\sqrt{2+\sqrt{5}}+\sqrt{3-\sqrt{7}}}} $$
Taking the first number, we can take the rational numbers as $F_0$, where $k_0=2$ and $k_1=\sqrt2$, where $k_1$ is not in $F_0$. Then:
$$ k_2=1+\sqrt2 \\ k_3=3+\sqrt{1+\sqrt2} \\ k_3=5+\sqrt{3+\sqrt{1+\sqrt2}} $$
which eventually gives $\sqrt{5+\sqrt{3+\sqrt{1+\sqrt2}}}$. But that doesn't match the initial number. Is it possible to take multiple $k$ values when extending a field? What steps do I need to take verify the above numbers?