
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 ellipse \\(C\\): \\( \dfrac {x^{2}}{a^{2}} + \dfrac {y^{2}}{b^{2}} = 1 (a > b > 0)\\) has an eccentricity of \\( \dfrac { \sqrt {3}}{2}\\), and the length of the major axis of ellipse \\(C\\) is \\(2\\).  
\\((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, please explain why.

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