
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 a moving point $P(x,y)$ and a fixed point $F(3\sqrt{3},0)$, the ratio of the distance from $P$ to $F$ and its distance to the line $l:x=4\sqrt{3}$ is a constant $\frac{\sqrt{3}}{2}$.  
$(1)$ Find the equation of the trajectory of the moving point $P$;  

$(2)$ If a line $l$ passing through the point $M(4,2)$ intersects the trajectory of the moving point $P$ at two distinct points $B$ and $C$, and $M$ is the midpoint of the line segment $BC$, find the equation of the line $l$.

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