
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 Cartesian coordinate system $xOy$, there are circles $C_{1}:x^{2}+y^{2}=4$ and $C_{2}:(x-2)^{2}+y^{2}=4$.

$(1)$ In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, write the polar equations for circles $C_{1}$ and $C_{2}$, respectively, and find the coordinates of the intersection points of circles $C_{1}$ and $C_{2}$ (expressed in polar coordinates);

$(2)$ Find the equation of the common chord of circles $C_{1}$ and $C_{2}$.

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