
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 the transformation $T$ corresponding to the matrix $A= \begin{bmatrix} 1 & 0 \\ 0 & \sqrt {2}\end{bmatrix}$ transforms the curve $C$ into the curve $C_{1}$: $\frac {x^{2}}{4}+ \frac {y^{2}}{2}=1$, find the equation of the curve $C$.

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