
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, whose left and right vertices are $A$ and $B$ respectively, and $|AB| = 4$, with an eccentricity of $\frac{1}{2}$.

(Ⅰ) Find the equation of the ellipse $C$;  
(Ⅱ) Let point $Q(4,0)$, if point $P$ is 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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