
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 the polar equations of circle $O_1$ and circle $O_2$ are $\rho=2$ and $\rho^2 - 2\sqrt{2}\rho\cos(\theta - \frac{\pi}{4}) = 2$, respectively.  
(1) Convert the polar equations of circle $O_1$ and circle $O_2$ into Cartesian coordinate equations;  
(2) Suppose the intersection points of the two circles are $A$ and $B$, find the parametric equation of line $AB$, and use the parametric equation of line $AB$ to calculate the length of the common chord $|AB|$.

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