
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 two 2nd-order matrices $A=\begin{bmatrix} 1 & a \\ b & 2 \end{bmatrix}$ and $B=\begin{bmatrix} -1 & 2 \\ \frac{1}{2} & 1 \end{bmatrix}$ that satisfy $AB=\begin{bmatrix} c & 2 \\ 3 & d \end{bmatrix}$, where $a,b,c,d \in R$.

1. Find the values of $a,b,c,d$.
2. If curve $C_{1}:2x^{2}-2xy+1=0$ is transformed into another curve $C_{2}$ under the transformation corresponding to matrix $A$, find the equation of $C_{2}$.

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