
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 that \\(A\\) and \\(B\\) are the left and right vertices of the ellipse \\(\Gamma\\): \\(\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)\\), and \\(|AB| = 4\\), with an eccentricity of \\(\dfrac{\sqrt{2}}{2}\\).

\\((1)\\) Find the equation of the ellipse \\(\Gamma\\).

\\((2)\\) If point \\(P(x_0, y_0) (y_0 \neq 0)\\) is any point on the line \\(x = 4\\), and \\(PA\\), \\(PB\\) intersect the ellipse \\(\Gamma\\) at points \\(C\\) and \\(D\\) respectively, determine whether the line \\(CD\\) always passes through a fixed point. If it does, find this fixed point; if not, explain why.

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