IR-TST 2015 Problem 7

$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$ . Prove that : $abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$