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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