
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)\\), the circle \\(C_1\\): \\(x^2 + y^2 = a^2 + b^2\\) is called the "companion circle" of the ellipse \\(C\\). It is known that the eccentricity of the ellipse \\(C\\) is \\(\dfrac{\sqrt{3}}{2}\\), and it passes through the point \\((0,1)\\).
\\((1)\\) Find the standard equation of the ellipse \\(C\\);
\\((2)\\) If a line \\(l\\) passing through the point \\(P(0,m) (m > 0)\\) has only one common point with the ellipse \\(C\\), and the chord length intercepted by the companion circle \\(C_1\\) of the ellipse \\(C\\) is \\(2\sqrt{2}\\), find the value of the real number \\(m\\).

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