\[\large{\left( \dfrac{h_a^t + h_b^t + h_c^t}{3} \right)^{1/t} \leq \dfrac{\sqrt{3}}{2}\left( \dfrac{a^t + b^t + c^t}{3}\right)^{1/t}}\]

Let \(a,b,c\) be the sides of a triangle and \(h_a, h_b, h_c\) respectively be the corresponding altitudes. If the maximum range of validity of the above inequality for \(t\) be: \(\quad \large{-\alpha < t < \alpha}\),

where \(t \neq 0\) and \(\alpha = \dfrac{\ln(A)}{\ln(B)}\), and where \(A,B \in \mathbb R\), find the value of \(\dfrac{A}{B}\).

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