
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
Elective 4-4: Coordinate System and Parametric Equations

Given the parametric equations of curve \\(C\\) as \\(\begin{cases}x=2\cos \left(\theta\right) \\ y= \sqrt{3}\sin \left(\theta\right)\end{cases} \\), in the same plane Cartesian coordinate system, the points on curve \\(C\\) are transformed by \\(\begin{cases} {x'}=\dfrac{1}{2}x \\ {y'}=\dfrac{1}{\sqrt{3}}y \\ \end{cases}\\) to obtain curve \\({C'}\\), with the origin as the pole and the positive half-axis of \\(x\\) as the polar axis, establishing a polar coordinate system.  
   \\((\\)Ⅰ\\()\\) Find the polar equation of curve \\({C'}\\);  
   \\((\\)Ⅱ\\()\\) If a line \\(l\\) passing through point \\(A\left(\dfrac{3}{2},\pi \right)\\) (in polar coordinates) with a slope angle of \\(\dfrac{\pi }{6}\\) intersects curve \\({C'}\\) at points \\(M\\) and \\(N\\), and the midpoint of chord \\(MN\\) is \\(P\\), find the value of \\(\dfrac{|AP|}{|AM|\cdot |AN|}\\).

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