
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 matrix $A = \left[ \begin{matrix} 1 & 0 \\ 0 & \sqrt{2} \end{matrix} \right]$ corresponds to the transformation $T$ that transforms curve $C$ into curve $C_1: \frac{x^2}{4} + \frac{y^2}{2} = 1$, then the equation of curve $C$ is __________.

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