
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 any plane vector $\overrightarrow {AB} = (x, y)$, rotating $\overrightarrow {AB}$ around its starting point in the counterclockwise direction by an angle $\theta$ results in vector $\overrightarrow {AP} = (x\cos\theta - y\sin\theta, x\sin\theta + y\cos\theta)$, which is equivalent to rotating point B around point A in the counterclockwise direction by an angle $\theta$ to obtain point P. If the trajectory of points obtained by rotating each point on curve C in the plane around the origin in the counterclockwise direction by $\frac {\pi }{4}$ is the curve $x^2 - y^2 = 2$, then the equation of the original curve C is $\boxed{xy = -1}$.

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