
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let  $x,y$  be real numbers such that  $xy=1$ . Let  $T$  and  $t$  be the largest and smallest values of the expression  $\hspace{2cm} \frac{(x+y)^2-(x-y)-2}{(x+y)^2+(x-y)-2}$ .

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

Present the answer in LaTex format: \boxed{Your answer}
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