
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 Cartesian coordinate system, with the origin _\\(O\\)_ as the pole and the positive half-axis of _\\(x\\)_ as the polar axis, the polar equation of curve _\\(C\\)_\\({\\,\\!}\_{1}\\) is _\\(ρ\\)_\\({\\,\\!}^{2}(1+3\\sin ^{2} \\)_\\(θ\\)_\\()=4.\\) Curve _\\(C\\)_\\({\\,\\!}\_{2}\\) is a circle with its center on the polar axis and passing through the pole. The ray _\\(θ\\)_\\(= \\dfrac{π}{3}\\) intersects curve _\\(C\\)_\\({\\,\\!}\_{2}\\) at point _\\(D\\)_\\(\\left(\\begin{matrix} \\begin{matrix}2, \\dfrac{π}{3} \\end{matrix}\\end{matrix}\\right)\\).

\\((1)\\) Find the Cartesian equations of curves _\\(C\\)_\\({\\,\\!}\_{1}\\) and _\\(C\\)_\\({\\,\\!}\_{2}\\);

\\((2)\\) Given two points in the polar coordinate system _\\(A\\)_\\((\\)_\\(ρ\\)_\\({\\,\\!}\_{1}\\),_\\(θ\\)_\\({\\,\\!}\_{0})\\) and _\\(B\\)_\\(\\left(\\begin{matrix} \\begin{matrix}ρ\_2 ,θ\_0 + \\dfrac{π}{2} \\end{matrix}\\end{matrix}\\right)\\), if _\\(A\\)_ and _\\(B\\)_ are both on curve _\\(C\\)_\\({\\,\\!}\_{1}\\), find the value of \\( \\dfrac{1}{ρ^{2}\_{1}}+ \\dfrac{1}{ρ^{2}\_{2}}\\).

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