
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 \(C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 \; (a > b > 0)\) with an eccentricity of \(\dfrac{\sqrt{2}}{2}\), the area of the triangle formed by any three vertices of ellipse \(C\) is \(2\sqrt{2}\).  
(I) Find the equation of ellipse \(C\);  
(II) Let \(A\) be the right vertex of ellipse \(C\), and point \(B\) is on the \(x\)-axis. If there exists a point \(P\) on ellipse \(C\) such that \(\angle APB = 90^\circ\), find the range of values for the \(x\)-coordinate of point \(B\).

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