
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
Given the inverse matrix $A^{-1} = \begin{bmatrix} \frac { \sqrt {2}}{2} & \frac { \sqrt {2}}{2} \\ - \frac { \sqrt {2}}{2} & \frac { \sqrt {2}}{2}\end{bmatrix}$, find the equation of the curve obtained by applying the transformation corresponding to matrix $A$ to the curve $xy=1$.

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