
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 polar coordinate system, the polar equation of curve $C$ is $\rho =6\sin \theta$, and the polar coordinates of point $P$ are $(\sqrt{2},\frac{\pi }{4})$. Taking the pole as the origin of coordinates and the positive half-axis of the $x$-axis as the polar axis, a plane rectangular coordinate system is established.

    (1) Find the rectangular coordinate equation of curve $C$ and the rectangular coordinates of point $P$;

    (2) A line $l$ passing through point $P$ intersects curve $C$ at points $A$ and $B$. If $|PA|=2|PB|$, find the value of $|AB|$.

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