
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\\): \\(\dfrac{x^{2}}{b^{2}} + \dfrac{y^{2}}{a^{2}} = 1 (a > b > 0)\\) with eccentricity \\(\dfrac{\sqrt{3}}{2}\\), and the major axis of ellipse \\(C\\) is \\(4\\).  
\\((1)\\) Find the equation of ellipse \\(C\\);  
\\((2)\\) It is known that the line \\(l\\): \\(y = kx + \sqrt{3}\\) intersects ellipse \\(C\\) at points \\(A\\) and \\(B\\). Is there a real number \\(k\\) such that the circle with diameter \\(AB\\) just passes through the origin \\(O\\)? If it exists, find the value of \\(k\\); if not, explain why.

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