SE-SKOLORNAS 1987 Problem 5

Show that there exists a positive number t such that for all positive numbers $a,b,c,d$ with $abcd = 1$ , $$\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}+\frac{1}{1+d}> t.$$ and find the largest $t$ with this property.