
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 $(xOy)$, given the line $l$: $x+\sqrt{3}y=5\sqrt{3}$, establish a polar coordinate system with the origin $O$ as the pole and the non-negative semi-axis of $x$ as the polar axis. The polar equation of circle $C$ is $\rho =4\sin \theta$.

(1) Find the polar equation of line $l$ and the rectangular equation of circle $C$;

(2) The ray $OP$: $\theta =\dfrac{\pi }{6}(\rho \geqslant 0)$ intersects circle $C$ at points $O$ and $A$, and intersects line $l$ at point $B$. Find the length of segment $AB$.

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