
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
In the elective course "4-4: Coordinate System and Parametric Equations", with the origin \\(O\\) as the pole and the positive half-axis of the \\(x\\)-axis as the polar axis, a polar coordinate system is established. It is known that the polar equation of curve \\(C\\) is \\({ρ}^{2}(1+3{\\sin }^{2}θ)=4 \\).

\\((\\)Ⅰ\\()\\) Find the parametric equation of curve \\(C\\);

\\((\\)Ⅱ\\()\\) If the curve intersects the positive half-axis of the \\(x\\)-axis and the positive half-axis of the \\(y\\)-axis at points \\(A\\) and \\(B\\) respectively, and a point \\(P\\) is arbitrarily chosen on curve \\(C\\) with \\(P\\) in the first quadrant, find the maximum area of quadrilateral \\(OAPB\\).

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