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

$(1)$ Find the equation of the trajectory $C$ of point $P$;

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

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