
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 \\(\dfrac{1}{2}\\).

\\((1)\\) Find the equation of the trajectory \\(E\\) of the moving point \\(P\\);

\\((2)\\) Suppose the trajectory \\(E\\) intersects the negative half-axis 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 the 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, please explain why.

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