
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]


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Given an ellipse \\(C\\) with its center at the origin of coordinates, its foci on the \\(x\\)-axis, and its eccentricity as \\(\frac{\sqrt{2}}{2}\\). A perpendicular line to the \\(x\\)-axis is drawn from a point \\(P(2,1)\\) on the ellipse \\(C\\), and the foot of the perpendicular is \\(Q\\).

\\((1)\\) Find the equation of the ellipse \\(C\\);

\\((2)\\) A line \\(l\\) passing through point \\(Q\\) intersects the ellipse \\(C\\) at points \\(A\\) and \\(B\\), and \\(3\overrightarrow{QA}+\overrightarrow{QB}=\overrightarrow{0}\\), find the equation of line \\(l\\).

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