For a positive integer $n$, let $A_n$ be the set of quadruplets $(a,b,c,d)$ of integers, satisfying the following properties simultaneously: $0\le a\le c\le n,$ $0\le b\le d\le n,$ $c+d>n,$ and $bc=ad+1.$ Moreover, define $$\alpha_n=\sum_{(a,b,c,d)\in A_n}\frac{1}{ab+cd}.$$Find all real numbers $t$ such that $\alpha_n>t$ for every positive integer $n$.