
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 that the polar coordinate 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) Find the rectangular coordinate equations of circle $O_1$ and circle $O_2$;

(2) Let the intersection points of the two circles be $A$ and $B$, find the length of the chord $\left| AB \right|$.

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