
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 the parametric equation of curve \\(C\\) as \\(\begin{cases} & x=2\cos \theta \\ & y=\sin \theta \end{cases}( \theta \\) is the parameter\\()), with the origin \\(O\\) as the pole and the positive half-axis of \\(x\\) as the polar axis, the polar equation of line \\(l\\) is \\(\rho \sin \left( \frac{\pi }{4}+\theta \right)=2\sqrt{2}\\).

\\((1)\\) Stretch the ordinate of each point on curve \\(C\\) to twice its original length to get curve \\({C_1}\\), write the polar equation of curve \\({C_1}\\);

\\((2)\\) The ray \\(\theta =\frac{\pi }{6}\\) intersects \\({C_1},l\\) at points \\(A,B\\) respectively, and the ray \\(\theta =-\frac{\pi }{6}\\) intersects \\({C_1},l\\) at points \\({A_1},{B_1}\\) respectively. Find the ratio of the areas of \\(\Delta OA{A_1}\\) and \\(\Delta OB{B_1}\\).

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