Let \( \triangle ABC \) be an isosceles triangle in which \( A \) is at \( (-1, 0) \), \( \angle A = \dfrac{2\pi}{3} \), \( AB = AC \) and \( B \) is on the positive \( x \)-axis. If \( BC = 4\sqrt{3} \) and the line \( BC \) intersects the line \( y = x + 3 \) at \( (\alpha, \beta) \) then \( \dfrac{\beta^{4}}{\alpha^{2}} \) is __________.
