# Inverse Fibonacci

$a = \displaystyle \sum_{i=1}^{\infty} \frac {F_i}{10^{i+1}},$

where $$F_i$$ are the Fibonacci numbers satisfying the relation $$F_{i+1}=F_i+F_{i-1}$$ with $$F_1=1,F_2=1$$.

Let $$\alpha,\beta$$ be the two (not necessarily distinct) solutions of the quadratic equation $$a^2x^2-2ax+1-a^2=0$$.

Find the value of $$\large{|\alpha^2-\beta^2|}$$.

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