
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 rectangular coordinate system $(xOy)$, there is a circle $C_{1}$: $(x-3)^{2}+y^{2}=9$. Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The polar coordinates of the center of another circle $C_{2}$ are $(\sqrt{2}, \frac{\pi}{4})$, with a radius of $1$.
(1) Find the polar equation of circle $C_{1}$;
(2) Suppose circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Find the length $|AB|$.

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