
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 $A$ is a moving point on the circle $P$: $(x-4)^{2}+y^{2}=36$, the coordinates of $B$ are $(-2,0)$, and $P$ is on the line segment $AB$, satisfying $\frac{|BP|}{|AP|}=\frac{1}{2}$.

(I) Find the equation of the trajectory $C$ of $P$.

(II) The line $l$ passing through the point $(-1,3)$ intersects $C$ at points $M$ and $N$, and $|MN|=2\sqrt{3}$, find the equation of the line $l$.

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