
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]


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Elective 4-2: Matrices and Transformations

Given matrices $A=\begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}$, $B=\begin{bmatrix} 1 & 0 \\ 0 & 2 \end{bmatrix}$.

$(1)$ Find $AB$;

$(2)$ If the curve $C_{1}: \frac{x^{2}}{8}+\frac{y^{2}}{2}=1$ is transformed by the matrix $AB$ into another curve $C_{2}$, find the equation of $C_{2}$.

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