
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 that for any planar vector $\overrightarrow{AB}=(x,y)$, rotating $\overrightarrow{AB}$ counterclockwise around its starting point by an angle $\theta$ yields the vector $\overrightarrow{AP}=(x\cos \theta-y\sin \theta,x\sin \theta+y\cos \theta)$, which is called rotating point $B$ counterclockwise around point $A$ by an angle $\theta$ to obtain point $P$. If each point on the planar curve $C$ is rotated counterclockwise around the origin by an angle of $\frac{\pi}{4}$ to obtain a point, and the trajectory of these points forms the curve $x^{2}-y^{2}=2$, what is the equation of the original curve $C$?

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