
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 4-4; Coordinate System and Parametric Equations

In the Cartesian coordinate system $xOy$, let line $C_{1}: x=-2$ and circle $C_{2}: (x-1)^{2}+(y-2)^{2}=1$. Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis.

$(1)$ Find the polar equations of $C_{1}$ and $C_{2}$.

$(2)$ If the polar equation of line $C_{3}$ is $\theta= \frac{\pi}{4} (\rho\in\mathbb{R})$, and the intersection points of $C_{2}$ and $C_{3}$ are $M$ and $N$, find the area of $\triangle C_{2}MN$.

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