
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
The point P$(x, y)$ is at a constant ratio of $\frac{\sqrt{3}}{2}$ between its distance to a fixed point F$(3\sqrt{3}, 0)$ and its distance to the line $l: x = 4\sqrt{3}$,  
(Ⅰ) Find the equation of the trajectory of point P;  
(Ⅱ) If a line m intersects the trajectory of P at two distinct points B and C, and the midpoint of segment BC is M$(4, 2)$, find the equation of line m.

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