Let the locus of the centre \( (\alpha, \beta) \), \( \beta > 0 \), of the circle which touches the circle \( x^2 + (y-1)^2 = 1 \) externally and also touches the \( x \)-axis be \( L \). Then the area bounded by \( L \) and the line \( y = 4 \) is:
\( \dfrac{32\sqrt{2}}{3} \)
\( \dfrac{40\sqrt{2}}{3} \)
\( \dfrac{64}{3} \)
\( \dfrac{32}{3} \)
