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Show that $$\sum_{n=1}^\infty \frac{z^{2^n}}{1-z^{2^{n+1}}}$$ is algebraic.

More specifically, solve this and get exact values.

Then use the result to evaluate $$\sum_{n=0}^\infty \frac{1}{F_{2^n}}$$ where $$F_n=\frac{\alpha^n-\beta^n}{\alpha-\beta}$$ and $\alpha=\frac{1+\sqrt{5}}{2}$ and $\beta=\frac{1-\sqrt{5}}{2}$.

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