Let $f(x)=\int \frac{d x}{\left(3+4 x^2\right) \sqrt{4-3 x^2}},|x|<\frac{2}{\sqrt{3}}$. If $f(0)=0$ and $f(1)=\frac{1}{\alpha \beta} \tan ^{-1}\left(\frac{\alpha}{\beta}\right), \alpha, \beta>0$, then $\alpha^2+\beta^2$ is equal to. $\_\_\_\_$ .
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