
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 an ellipse \\(C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)\\) with its left and right foci being \\(F_1\\) and \\(F_2\\) respectively, and its eccentricity \\(e = \frac{\sqrt{2}}{2}\\). A line passing through \\(F_2\\) and perpendicular to the x-axis intersects the ellipse \\(C\\) at point \\(P\\) in the first quadrant, and \\(|OP| = \frac{\sqrt{6}}{2}\\).
\\((1)\\) Find the equation of the ellipse \\(C\\);
\\((2)\\) A line \\(l\\) passing through point \\(Q(0,2)\\) intersects the ellipse \\(C\\) at points \\(A\\) and \\(B\\). When the area of triangle \\(AOB\\) is \\(\frac{\sqrt{2}}{2}\\), find the equation of line \\(l\\).

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