Let $x,y$ be real numbers such that $xy=1$. Let $T$ and $t$ be the largest and smallest values of the expression:

$$ \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2} $$

If $T+t$ can be expressed in the form $\frac{m}{n}$ where $m,n$ are nonzero integers with $\text{GCD}(m,n)=1$, find the value of $m+n$.

Let's think step by step and output the final answer within \boxed{}.
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