
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
In the plane rectangular coordinate system $xOy$, a curve $C$ is first transformed by matrix $A=\begin{bmatrix} \cos{\theta} & -\sin{\theta} \\ \sin{\theta} & \cos{\theta} \end{bmatrix}$ ($0<\theta<2\pi$), and then the resulting curve is transformed by matrix $B=\begin{bmatrix} 1 & 0 \\ 0 & k \end{bmatrix}$ ($0<k<1$). If the matrix corresponding to the continuous implementation of these two transformations is $\begin{bmatrix} 0 & -1 \\ \frac{1}{2} & 0 \end{bmatrix}$, find the values of $k$ and $\theta$.

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