
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: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$), where the left and right vertices are A and B, respectively, and $|AB| = 4$. The eccentricity of the ellipse is $\frac{1}{2}$.  
(Ⅰ) Find the equation of ellipse C;  
(Ⅱ) Let point Q be (4, 0). If point P lies on the line $x = 4$, and line BP intersects the ellipse at another point M. Determine whether there exists a point P such that quadrilateral APQM is a trapezoid? If it exists, find the coordinates of point P; if not, explain why.

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