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

(Ⅰ) Find the equation of the trajectory $E$ of the moving point $P$;  
(Ⅱ) Suppose the trajectory $E$ intersects the negative half of the x-axis at point $A$, and a line through point $F$ that does not coincide with the x-axis intersects the trajectory $E$ at two points $B$ and $C$. The lines $AB$ and $AC$ intersect line $l$ at points $M$ and $N$, respectively. Is there a fixed point $Q$ on the x-axis such that $\overrightarrow{QM} \cdot \overrightarrow{QN} = 0$? If it exists, find the coordinates of the fixed point $Q$; if not, explain why.

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