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