MEMO 2007 T1

Let $a,b,c,d$ be real numbers which satisfy $\frac{1}{2}\leq a,b,c,d\leq 2$ and $abcd=1$ . Find the maximum value of $$\left(a+\frac{1}{b}\right)\left(b+\frac{1}{c}\right)\left(c+\frac{1}{d}\right)\left(d+\frac{1}{a}\right).$$