Maximum? This shouldn't be tough

Algebra Level 5

Let a,b,c,d[12,2]a, b, c, d \in \left[\frac{1}{2}, 2\right]. Suppose abcd=1abcd = 1. Find the maximum possible value of (a+1b)(b+1c)(c+1d)(d+1a)\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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