
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): $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with an eccentricity of $\frac{\sqrt{3}}{2}$. The ellipse intersects the y-axis at points A and B, with |AB| = 2.
(I) Find the standard equation of the ellipse (C);
(II) Let P be a moving point on the ellipse (C) that is on the right side of the y-axis. Lines PA and PB intersect with the line x = 4 at points M and N, respectively. A circle with diameter MN intersects the x-axis at points E and F. Find the range of the x-coordinate of point P and the maximum value of |EF|.

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