
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 a curve $C$ undergoes a transformation corresponding to the matrix $M$ $(=\begin{bmatrix}1 & 0 \\ 0 & 2\end{bmatrix})$ to produce curve $C_1$, and $C_1$ undergoes a transformation corresponding to the matrix $N$ $(=\begin{bmatrix}0 & -1 \\ 1 & 0\end{bmatrix})$ to produce curve $C_2$. If the equation of curve $C_2$ is $y=\frac{1}{8}x^2$, find the equation of curve $C$.

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