Let $a, b, c, d \in \left[\frac{1}{2}, 2\right]$. Suppose $abcd = 1$. Find the maximum possible value of $\left(a + \dfrac{1}{b}\right)\left(b + \dfrac{1}{c}\right)\left(c + \dfrac{1}{d}\right)\left(d + \dfrac{1}{a}\right)$.

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